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zwischen ",[572,2303,2305,2322],{"className":2304},[575],[572,2306,2308],{"className":2307},[579],[581,2309,2310],{"xmlns":583},[585,2311,2312,2319],{},[588,2313,2314,2317],{},[678,2315,2316],{},"−",[1150,2318,1152],{},[602,2320,2321],{"encoding":604},"-1",[572,2323,2325],{"className":2324,"ariaHidden":610},[609],[572,2326,2328,2332,2335],{"className":2327},[614],[572,2329],{"className":2330,"style":2331},[618],"height:0.7278em;vertical-align:-0.0833em;",[572,2333,2316],{"className":2334},[623],[572,2336,1152],{"className":2337},[623],[572,2339,2341,2354],{"className":2340},[575],[572,2342,2344],{"className":2343},[579],[581,2345,2346],{"xmlns":583},[585,2347,2348,2352],{},[588,2349,2350],{},[1150,2351,1152],{},[602,2353,1152],{"encoding":604},[572,2355,2357],{"className":2356,"ariaHidden":610},[609],[572,2358,2360,2364],{"className":2359},[614],[572,2361],{"className":2362,"style":2363},[618],"height:0.6444em;",[572,2365,1152],{"className":2366},[623],". Erst dadurch ist der ",[568,2369,2370],{},"Winkel"," zwischen zwei Vektoren über den Kosinus wohldefiniert.",[1768,2373,566,2374,2377,2378,2650,2651,891],{},[568,2375,2376],{},"Dreiecksungleichung"," 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folgt durch Ausmultiplizieren von ",[572,2652,2654,2688],{"className":2653},[575],[572,2655,2657],{"className":2656},[579],[581,2658,2659],{"xmlns":583},[585,2660,2661,2685],{},[588,2662,2663,2665,2671,2673,2679],{},[678,2664,2030],{"stretchy":680},[911,2666,2667,2669],{"accent":610},[594,2668,915],{},[678,2670,918],{},[678,2672,2400],{},[911,2674,2675,2677],{"accent":610},[594,2676,1000],{},[678,2678,918],{},[591,2680,2681,2683],{},[678,2682,2030],{"stretchy":680},[1150,2684,1172],{},[602,2686,2687],{"encoding":604},"\\lVert \\vec u + \\vec v \\rVert^2",[572,2689,2691,2744],{"className":2690,"ariaHidden":610},[609],[572,2692,2694,2697,2700,2735,2738,2741],{"className":2693},[614],[572,2695],{"className":2696,"style":749},[618],[572,2698,2030],{"className":2699},[753],[572,2701,2703],{"className":2702},[623,935],[572,2704,2706],{"className":2705},[635],[572,2707,2709],{"className":2708},[639],[572,2710,2712,2720],{"className":2711,"style":931},[643],[572,2713,2714,2717],{"style":947},[572,2715],{"className":2716,"style":951},[651],[572,2718,915],{"className":2719},[623,663],[572,2721,2722,2725],{"style":947},[572,2723],{"className":2724,"style":951},[651],[572,2726,2728],{"className":2727,"style":964},[963],[572,2729,2731],{"className":2730,"style":969},[968],[971,2732,2733],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,2734],{"d":982},[572,2736],{"className":2737,"style":2486},[765],[572,2739,2400],{"className":2740},[2490],[572,2742],{"className":2743,"style":2486},[765],[572,2745,2747,2751,2786],{"className":2746},[614],[572,2748],{"className":2749,"style":2750},[618],"height:1.0641em;vertical-align:-0.25em;",[572,2752,2754],{"className":2753},[623,935],[572,2755,2757],{"className":2756},[635],[572,2758,2760],{"className":2759},[639],[572,2761,2763,2771],{"className":2762,"style":931},[643],[572,2764,2765,2768],{"style":947},[572,2766],{"className":2767,"style":951},[651],[572,2769,1000],{"className":2770,"style":770},[623,663],[572,2772,2773,2776],{"style":947},[572,2774],{"className":2775,"style":951},[651],[572,2777,2779],{"className":2778,"style":964},[963],[572,2780,2782],{"className":2781,"style":969},[968],[971,2783,2784],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,2785],{"d":982},[572,2787,2789,2792],{"className":2788},[774],[572,2790,2030],{"className":2791},[774],[572,2793,2795],{"className":2794},[631],[572,2796,2798],{"className":2797},[635],[572,2799,2801],{"className":2800},[639],[572,2802,2805],{"className":2803,"style":2804},[643],"height:0.8141em;",[572,2806,2807,2810],{"style":647},[572,2808],{"className":2809,"style":652},[651],[572,2811,2813],{"className":2812},[656,657,658,659],[572,2814,1172],{"className":2815},[623,659],[564,2817,2818,2819,2975,2976,3168,3169,3198,3199,3227,3228,3272],{},"Beweisidee: Für ",[572,2820,2822,2848],{"className":2821},[575],[572,2823,2825],{"className":2824},[579],[581,2826,2827],{"xmlns":583},[585,2828,2829,2845],{},[588,2830,2831,2837,2839],{},[911,2832,2833,2835],{"accent":610},[594,2834,1000],{},[678,2836,918],{},[678,2838,1799],{"mathvariant":1234},[911,2840,2841,2843],{"accent":610},[1150,2842,1804],{},[678,2844,918],{},[602,2846,2847],{"encoding":604},"\\vec v \\neq \\vec 0",[572,2849,2851,2934],{"className":2850,"ariaHidden":610},[609],[572,2852,2854,2857,2892,2895,2931],{"className":2853},[614],[572,2855],{"className":2856,"style":1819},[618],[572,2858,2860],{"className":2859},[623,935],[572,2861,2863],{"className":2862},[635],[572,2864,2866],{"className":2865},[639],[572,2867,2869,2877],{"className":2868,"style":931},[643],[572,2870,2871,2874],{"style":947},[572,2872],{"className":2873,"style":951},[651],[572,2875,1000],{"className":2876,"style":770},[623,663],[572,2878,2879,2882],{"style":947},[572,2880],{"className":2881,"style":951},[651],[572,2883,2885],{"className":2884,"style":964},[963],[572,2886,2888],{"className":2887,"style":969},[968],[971,2889,2890],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,2891],{"d":982},[572,2893],{"className":2894,"style":778},[765],[572,2896,2898,2925,2928],{"className":2897},[782],[572,2899,2901],{"className":2900},[782],[572,2902,2904],{"className":2903},[623,1908],[572,2905,2907],{"className":2906},[1912],[572,2908,2910,2913,2922],{"className":2909},[1916],[572,2911],{"className":2912,"style":1920},[618],[572,2914,2916],{"className":2915},[1924],[572,2917,2919],{"className":2918},[623],[572,2920,1931],{"className":2921},[782],[572,2923],{"className":2924},[1935],[572,2926],{"className":2927},[765,1939],[572,2929,696],{"className":2930},[782],[572,2932],{"className":2933,"style":778},[765],[572,2935,2937,2940],{"className":2936},[614],[572,2938],{"className":2939,"style":1952},[618],[572,2941,2943],{"className":2942},[623,935],[572,2944,2946],{"className":2945},[635],[572,2947,2949],{"className":2948},[639],[572,2950,2952,2960],{"className":2951,"style":1952},[643],[572,2953,2954,2957],{"style":947},[572,2955],{"className":2956,"style":951},[651],[572,2958,1804],{"className":2959},[623],[572,2961,2962,2965],{"style":1975},[572,2963],{"className":2964,"style":951},[651],[572,2966,2968],{"className":2967,"style":1982},[963],[572,2969,2971],{"className":2970,"style":969},[968],[971,2972,2973],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,2974],{"d":982}," ist ",[572,2977,2979,3021],{"className":2978},[575],[572,2980,2982],{"className":2981},[579],[581,2983,2984],{"xmlns":583},[585,2985,2986,3018],{},[588,2987,2988,2990,2996,2998,3001,3007,3013,3016],{},[678,2989,2030],{"stretchy":680},[911,2991,2992,2994],{"accent":610},[594,2993,915],{},[678,2995,918],{},[678,2997,2316],{},[594,2999,3000],{},"t",[911,3002,3003,3005],{"accent":610},[594,3004,1000],{},[678,3006,918],{},[591,3008,3009,3011],{},[678,3010,2030],{"stretchy":680},[1150,3012,1172],{},[678,3014,3015],{},"≥",[1150,3017,1804],{},[602,3019,3020],{"encoding":604},"\\lVert \\vec u - t\\vec v \\rVert^2 \\ge 0",[572,3022,3024,3077,3159],{"className":3023,"ariaHidden":610},[609],[572,3025,3027,3030,3033,3068,3071,3074],{"className":3026},[614],[572,3028],{"className":3029,"style":749},[618],[572,3031,2030],{"className":3032},[753],[572,3034,3036],{"className":3035},[623,935],[572,3037,3039],{"className":3038},[635],[572,3040,3042],{"className":3041},[639],[572,3043,3045,3053],{"className":3044,"style":931},[643],[572,3046,3047,3050],{"style":947},[572,3048],{"className":3049,"style":951},[651],[572,3051,915],{"className":3052},[623,663],[572,3054,3055,3058],{"style":947},[572,3056],{"className":3057,"style":951},[651],[572,3059,3061],{"className":3060,"style":964},[963],[572,3062,3064],{"className":3063,"style":969},[968],[971,3065,3066],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3067],{"d":982},[572,3069],{"className":3070,"style":2486},[765],[572,3072,2316],{"className":3073},[2490],[572,3075],{"className":3076,"style":2486},[765],[572,3078,3080,3083,3086,3121,3150,3153,3156],{"className":3079},[614],[572,3081],{"className":3082,"style":2750},[618],[572,3084,3000],{"className":3085},[623,663],[572,3087,3089],{"className":3088},[623,935],[572,3090,3092],{"className":3091},[635],[572,3093,3095],{"className":3094},[639],[572,3096,3098,3106],{"className":3097,"style":931},[643],[572,3099,3100,3103],{"style":947},[572,3101],{"className":3102,"style":951},[651],[572,3104,1000],{"className":3105,"style":770},[623,663],[572,3107,3108,3111],{"style":947},[572,3109],{"className":3110,"style":951},[651],[572,3112,3114],{"className":3113,"style":964},[963],[572,3115,3117],{"className":3116,"style":969},[968],[971,3118,3119],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3120],{"d":982},[572,3122,3124,3127],{"className":3123},[774],[572,3125,2030],{"className":3126},[774],[572,3128,3130],{"className":3129},[631],[572,3131,3133],{"className":3132},[635],[572,3134,3136],{"className":3135},[639],[572,3137,3139],{"className":3138,"style":2804},[643],[572,3140,3141,3144],{"style":647},[572,3142],{"className":3143,"style":652},[651],[572,3145,3147],{"className":3146},[656,657,658,659],[572,3148,1172],{"className":3149},[623,659],[572,3151],{"className":3152,"style":778},[765],[572,3154,3015],{"className":3155},[782],[572,3157],{"className":3158,"style":778},[765],[572,3160,3162,3165],{"className":3161},[614],[572,3163],{"className":3164,"style":2363},[618],[572,3166,1804],{"className":3167},[623]," für alle ",[572,3170,3172,3185],{"className":3171},[575],[572,3173,3175],{"className":3174},[579],[581,3176,3177],{"xmlns":583},[585,3178,3179,3183],{},[588,3180,3181],{},[594,3182,3000],{},[602,3184,3000],{"encoding":604},[572,3186,3188],{"className":3187,"ariaHidden":610},[609],[572,3189,3191,3195],{"className":3190},[614],[572,3192],{"className":3193,"style":3194},[618],"height:0.6151em;",[572,3196,3000],{"className":3197},[623,663],". Das ist eine quadratische Funktion in ",[572,3200,3202,3215],{"className":3201},[575],[572,3203,3205],{"className":3204},[579],[581,3206,3207],{"xmlns":583},[585,3208,3209,3213],{},[588,3210,3211],{},[594,3212,3000],{},[602,3214,3000],{"encoding":604},[572,3216,3218],{"className":3217,"ariaHidden":610},[609],[572,3219,3221,3224],{"className":3220},[614],[572,3222],{"className":3223,"style":3194},[618],[572,3225,3000],{"className":3226},[623,663]," ohne zwei Nullstellen, ihre Diskriminante ist also ",[572,3229,3231,3247],{"className":3230},[575],[572,3232,3234],{"className":3233},[579],[581,3235,3236],{"xmlns":583},[585,3237,3238,3244],{},[588,3239,3240,3242],{},[678,3241,1175],{},[1150,3243,1804],{},[602,3245,3246],{"encoding":604},"\\le 0",[572,3248,3250,3263],{"className":3249,"ariaHidden":610},[609],[572,3251,3253,3257,3260],{"className":3252},[614],[572,3254],{"className":3255,"style":3256},[618],"height:0.7719em;vertical-align:-0.136em;",[572,3258,1175],{"className":3259},[782],[572,3261],{"className":3262,"style":778},[765],[572,3264,3266,3269],{"className":3265},[614],[572,3267],{"className":3268,"style":2363},[618],[572,3270,1804],{"className":3271},[623],". Ausgeschrieben ist das die Ungleichung.",[559,3274,3276],{"id":3275},"beispiele","Beispiele",[564,3278,3279,983,3413,3548,3549,3763,3764,3763,3861,3959,3960,4013,4014,891],{},[572,3280,3282,3321],{"className":3281},[575],[572,3283,3285],{"className":3284},[579],[581,3286,3287],{"xmlns":583},[585,3288,3289,3318],{},[588,3290,3291,3297,3299,3301,3303,3305,3308,3310,3312,3314,3316],{},[911,3292,3293,3295],{"accent":610},[594,3294,915],{},[678,3296,918],{},[678,3298,696],{},[678,3300,713],{"stretchy":680},[1150,3302,1152],{},[678,3304,687],{"separator":610},[729,3306,3307],{}," ",[1150,3309,1172],{},[678,3311,687],{"separator":610},[729,3313,3307],{},[1150,3315,1172],{},[678,3317,719],{"stretchy":680},[602,3319,3320],{"encoding":604},"\\vec u = (1,\\ 2,\\ 2)",[572,3322,3324,3374],{"className":3323,"ariaHidden":610},[609],[572,3325,3327,3330,3365,3368,3371],{"className":3326},[614],[572,3328],{"className":3329,"style":931},[618],[572,3331,3333],{"className":3332},[623,935],[572,3334,3336],{"className":3335},[635],[572,3337,3339],{"className":3338},[639],[572,3340,3342,3350],{"className":3341,"style":931},[643],[572,3343,3344,3347],{"style":947},[572,3345],{"className":3346,"style":951},[651],[572,3348,915],{"className":3349},[623,663],[572,3351,3352,3355],{"style":947},[572,3353],{"className":3354,"style":951},[651],[572,3356,3358],{"className":3357,"style":964},[963],[572,3359,3361],{"className":3360,"style":969},[968],[971,3362,3363],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3364],{"d":982},[572,3366],{"className":3367,"style":778},[765],[572,3369,696],{"className":3370},[782],[572,3372],{"className":3373,"style":778},[765],[572,3375,3377,3380,3383,3386,3389,3392,3395,3398,3401,3404,3407,3410],{"className":3376},[614],[572,3378],{"className":3379,"style":749},[618],[572,3381,713],{"className":3382},[753],[572,3384,1152],{"className":3385},[623],[572,3387,687],{"className":3388},[761],[572,3390,3307],{"className":3391},[765],[572,3393],{"className":3394,"style":766},[765],[572,3396,1172],{"className":3397},[623],[572,3399,687],{"className":3400},[761],[572,3402,3307],{"className":3403},[765],[572,3405],{"className":3406,"style":766},[765],[572,3408,1172],{"className":3409},[623],[572,3411,719],{"className":3412},[774],[572,3414,3416,3456],{"className":3415},[575],[572,3417,3419],{"className":3418},[579],[581,3420,3421],{"xmlns":583},[585,3422,3423,3453],{},[588,3424,3425,3431,3433,3435,3438,3440,3442,3444,3446,3448,3451],{},[911,3426,3427,3429],{"accent":610},[594,3428,1000],{},[678,3430,918],{},[678,3432,696],{},[678,3434,713],{"stretchy":680},[1150,3436,3437],{},"3",[678,3439,687],{"separator":610},[729,3441,3307],{},[1150,3443,1804],{},[678,3445,687],{"separator":610},[729,3447,3307],{},[1150,3449,3450],{},"4",[678,3452,719],{"stretchy":680},[602,3454,3455],{"encoding":604},"\\vec v = (3,\\ 0,\\ 4)",[572,3457,3459,3509],{"className":3458,"ariaHidden":610},[609],[572,3460,3462,3465,3500,3503,3506],{"className":3461},[614],[572,3463],{"className":3464,"style":931},[618],[572,3466,3468],{"className":3467},[623,935],[572,3469,3471],{"className":3470},[635],[572,3472,3474],{"className":3473},[639],[572,3475,3477,3485],{"className":3476,"style":931},[643],[572,3478,3479,3482],{"style":947},[572,3480],{"className":3481,"style":951},[651],[572,3483,1000],{"className":3484,"style":770},[623,663],[572,3486,3487,3490],{"style":947},[572,3488],{"className":3489,"style":951},[651],[572,3491,3493],{"className":3492,"style":964},[963],[572,3494,3496],{"className":3495,"style":969},[968],[971,3497,3498],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3499],{"d":982},[572,3501],{"className":3502,"style":778},[765],[572,3504,696],{"className":3505},[782],[572,3507],{"className":3508,"style":778},[765],[572,3510,3512,3515,3518,3521,3524,3527,3530,3533,3536,3539,3542,3545],{"className":3511},[614],[572,3513],{"className":3514,"style":749},[618],[572,3516,713],{"className":3517},[753],[572,3519,3437],{"className":3520},[623],[572,3522,687],{"className":3523},[761],[572,3525,3307],{"className":3526},[765],[572,3528],{"className":3529,"style":766},[765],[572,3531,1804],{"className":3532},[623],[572,3534,687],{"className":3535},[761],[572,3537,3307],{"className":3538},[765],[572,3540],{"className":3541,"style":766},[765],[572,3543,3450],{"className":3544},[623],[572,3546,719],{"className":3547},[774],": ",[572,3550,3552,3600],{"className":3551},[575],[572,3553,3555],{"className":3554},[579],[581,3556,3557],{"xmlns":583},[585,3558,3559,3597],{},[588,3560,3561,3563,3569,3571,3577,3579,3581,3583,3585,3587,3589,3592,3594],{},[678,3562,681],{"stretchy":680},[911,3564,3565,3567],{"accent":610},[594,3566,915],{},[678,3568,918],{},[678,3570,687],{"separator":610},[911,3572,3573,3575],{"accent":610},[594,3574,1000],{},[678,3576,918],{},[678,3578,693],{"stretchy":680},[678,3580,696],{},[1150,3582,3437],{},[678,3584,2400],{},[1150,3586,1804],{},[678,3588,2400],{},[1150,3590,3591],{},"8",[678,3593,696],{},[1150,3595,3596],{},"11",[602,3598,3599],{"encoding":604},"\\langle \\vec u, \\vec v \\rangle = 3 + 0 + 8 = 11",[572,3601,3603,3700,3718,3736,3754],{"className":3602,"ariaHidden":610},[609],[572,3604,3606,3609,3612,3647,3650,3653,3688,3691,3694,3697],{"className":3605},[614],[572,3607],{"className":3608,"style":749},[618],[572,3610,681],{"className":3611},[753],[572,3613,3615],{"className":3614},[623,935],[572,3616,3618],{"className":3617},[635],[572,3619,3621],{"className":3620},[639],[572,3622,3624,3632],{"className":3623,"style":931},[643],[572,3625,3626,3629],{"style":947},[572,3627],{"className":3628,"style":951},[651],[572,3630,915],{"className":3631},[623,663],[572,3633,3634,3637],{"style":947},[572,3635],{"className":3636,"style":951},[651],[572,3638,3640],{"className":3639,"style":964},[963],[572,3641,3643],{"className":3642,"style":969},[968],[971,3644,3645],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3646],{"d":982},[572,3648,687],{"className":3649},[761],[572,3651],{"className":3652,"style":766},[765],[572,3654,3656],{"className":3655},[623,935],[572,3657,3659],{"className":3658},[635],[572,3660,3662],{"className":3661},[639],[572,3663,3665,3673],{"className":3664,"style":931},[643],[572,3666,3667,3670],{"style":947},[572,3668],{"className":3669,"style":951},[651],[572,3671,1000],{"className":3672,"style":770},[623,663],[572,3674,3675,3678],{"style":947},[572,3676],{"className":3677,"style":951},[651],[572,3679,3681],{"className":3680,"style":964},[963],[572,3682,3684],{"className":3683,"style":969},[968],[971,3685,3686],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,3687],{"d":982},[572,3689,693],{"className":3690},[774],[572,3692],{"className":3693,"style":778},[765],[572,3695,696],{"className":3696},[782],[572,3698],{"className":3699,"style":778},[765],[572,3701,3703,3706,3709,3712,3715],{"className":3702},[614],[572,3704],{"className":3705,"style":2331},[618],[572,3707,3437],{"className":3708},[623],[572,3710],{"className":3711,"style":2486},[765],[572,3713,2400],{"className":3714},[2490],[572,3716],{"className":3717,"style":2486},[765],[572,3719,3721,3724,3727,3730,3733],{"className":3720},[614],[572,3722],{"className":3723,"style":2331},[618],[572,3725,1804],{"className":3726},[623],[572,3728],{"className":3729,"style":2486},[765],[572,3731,2400],{"className":3732},[2490],[572,3734],{"className":3735,"style":2486},[765],[572,3737,3739,3742,3745,3748,3751],{"className":3738},[614],[572,3740],{"className":3741,"style":2363},[618],[572,3743,3591],{"className":3744},[623],[572,3746],{"className":3747,"style":778},[765],[572,3749,696],{"className":3750},[782],[572,3752],{"className":3753,"style":778},[765],[572,3755,3757,3760],{"className":3756},[614],[572,3758],{"className":3759,"style":2363},[618],[572,3761,3596],{"className":3762},[623],", ",[572,3765,3767,3793],{"className":3766},[575],[572,3768,3770],{"className":3769},[579],[581,3771,3772],{"xmlns":583},[585,3773,3774,3790],{},[588,3775,3776,3778,3784,3786,3788],{},[678,3777,2030],{"stretchy":680},[911,3779,3780,3782],{"accent":610},[594,3781,915],{},[678,3783,918],{},[678,3785,2030],{"stretchy":680},[678,3787,696],{},[1150,3789,3437],{},[602,3791,3792],{"encoding":604},"\\lVert \\vec u \\rVert = 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Es gilt ",[572,3961,3963,3982],{"className":3962},[575],[572,3964,3966],{"className":3965},[579],[581,3967,3968],{"xmlns":583},[585,3969,3970,3979],{},[588,3971,3972,3974,3976],{},[1150,3973,3596],{},[678,3975,1175],{},[1150,3977,3978],{},"15",[602,3980,3981],{"encoding":604},"11 \\le 15",[572,3983,3985,4004],{"className":3984,"ariaHidden":610},[609],[572,3986,3988,3992,3995,3998,4001],{"className":3987},[614],[572,3989],{"className":3990,"style":3991},[618],"height:0.7804em;vertical-align:-0.136em;",[572,3993,3596],{"className":3994},[623],[572,3996],{"className":3997,"style":778},[765],[572,3999,1175],{"className":4000},[782],[572,4002],{"className":4003,"style":778},[765],[572,4005,4007,4010],{"className":4006},[614],[572,4008],{"className":4009,"style":2363},[618],[572,4011,3978],{"className":4012},[623],". Der Winkel ist ",[572,4015,4017,4052],{"className":4016},[575],[572,4018,4020],{"className":4019},[579],[581,4021,4022],{"xmlns":583},[585,4023,4024,4049],{},[588,4025,4026,4029,4032,4038,4041],{},[594,4027,4028],{},"arccos",[678,4030,4031],{},"⁡",[2004,4033,4034,4036],{},[1150,4035,3596],{},[1150,4037,3978],{},[678,4039,4040],{},"≈",[591,4042,4043,4046],{},[1150,4044,4045],{},"42,8",[678,4047,4048],{},"∘",[602,4050,4051],{"encoding":604},"\\arccos\\frac{11}{15} \\approx 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x\\,dx\\right)^2 = \\frac14 \\le \\int_0^1 x^2\\,dx \\cdot \\int_0^1 1\\,dx = 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5483],{"className":5484,"style":766},[765],[572,5486,734],{"className":5487},[623,663],[572,5489,716],{"className":5490},[623,663],[572,5492],{"className":5493,"style":2486},[765],[572,5495,5151],{"className":5496},[2490],[572,5498],{"className":5499,"style":2486},[765],[572,5501,5503,5506,5557,5560,5563,5566,5569,5572,5575,5578],{"className":5502},[614],[572,5504],{"className":5505,"style":5398},[618],[572,5507,5509,5512],{"className":5508},[796],[572,5510,702],{"className":5511,"style":802},[796,800,801],[572,5513,5515],{"className":5514},[631],[572,5516,5518,5549],{"className":5517},[635,809],[572,5519,5521,5546],{"className":5520},[639],[572,5522,5524,5535],{"className":5523,"style":5220},[643],[572,5525,5526,5529],{"style":819},[572,5527],{"className":5528,"style":652},[651],[572,5530,5532],{"className":5531},[656,657,658,659],[572,5533,1804],{"className":5534},[623,659],[572,5536,5537,5540],{"style":831},[572,5538],{"className":5539,"style":652},[651],[572,5541,5543],{"className":5542},[656,657,658,659],[572,5544,1152],{"className":5545},[623,659],[572,5547,845],{"className":5548},[844],[572,5550,5552],{"className":5551},[639],[572,5553,5555],{"className":5554,"style":852},[643],[572,5556],{},[572,5558],{"className":5559,"style":766},[765],[572,5561,1152],{"className":5562},[623],[572,5564],{"className":5565,"style":766},[765],[572,5567,734],{"className":5568},[623,663],[572,5570,716],{"className":5571},[623,663],[572,5573],{"className":5574,"style":778},[765],[572,5576,696],{"className":5577},[782],[572,5579],{"className":5580,"style":778},[765],[572,5582,5584,5587],{"className":5583},[614],[572,5585],{"className":5586,"style":4061},[618],[572,5588,5590,5593,5652],{"className":5589},[623],[572,5591],{"className":5592},[753,2070],[572,5594,5596],{"className":5595},[2004],[572,5597,5599,5644],{"className":5598},[635,809],[572,5600,5602,5641],{"className":5601},[639],[572,5603,5605,5619,5627],{"className":5604,"style":4086},[643],[572,5606,5607,5610],{"style":2086},[572,5608],{"className":5609,"style":951},[651],[572,5611,5613],{"className":5612},[656,657,658,659],[572,5614,5616],{"className":5615},[623,659],[572,5617,3437],{"className":5618},[623,659],[572,5620,5621,5624],{"style":2186},[572,5622],{"className":5623,"style":951},[651],[572,5625],{"className":5626,"style":2194},[2193],[572,5628,5629,5632],{"style":4111},[572,5630],{"className":5631,"style":951},[651],[572,5633,5635],{"className":5634},[656,657,658,659],[572,5636,5638],{"className":5637},[623,659],[572,5639,1152],{"className":5640},[623,659],[572,5642,845],{"className":5643},[844],[572,5645,5647],{"className":5646},[639],[572,5648,5650],{"className":5649,"style":4133},[643],[572,5651],{},[572,5653],{"className":5654},[774,2070],[559,5656,5658],{"id":5657},"typische-fehler","Typische Fehler",[1765,5660,5661,5913,6254],{},[1768,5662,5663,5664,5912],{},"Den Betrag links weglassen und dann nur ",[572,5665,5667,5721],{"className":5666},[575],[572,5668,5670],{"className":5669},[579],[581,5671,5672],{"xmlns":583},[585,5673,5674,5718],{},[588,5675,5676,5678,5684,5686,5692,5694,5696,5698,5704,5706,5708,5710,5716],{},[678,5677,681],{"stretchy":680},[911,5679,5680,5682],{"accent":610},[594,5681,915],{},[678,5683,918],{},[678,5685,687],{"separator":610},[911,5687,5688,5690],{"accent":610},[594,5689,1000],{},[678,5691,918],{},[678,5693,693],{"stretchy":680},[678,5695,1175],{},[678,5697,2030],{"stretchy":680},[911,5699,5700,5702],{"accent":610},[594,5701,915],{},[678,5703,918],{},[678,5705,2030],{"stretchy":680},[729,5707,731],{},[678,5709,2030],{"stretchy":680},[911,5711,5712,5714],{"accent":610},[594,5713,1000],{},[678,5715,918],{},[678,5717,2030],{"stretchy":680},[602,5719,5720],{"encoding":604},"\\langle \\vec u, \\vec v \\rangle \\le \\lVert \\vec u \\rVert\\,\\lVert \\vec v \\rVert",[572,5722,5724,5821],{"className":5723,"ariaHidden":610},[609],[572,5725,5727,5730,5733,5768,5771,5774,5809,5812,5815,5818],{"className":5726},[614],[572,5728],{"className":5729,"style":749},[618],[572,5731,681],{"className":5732},[753],[572,5734,5736],{"className":5735},[623,935],[572,5737,5739],{"className":5738},[635],[572,5740,5742],{"className":5741},[639],[572,5743,5745,5753],{"className":5744,"style":931},[643],[572,5746,5747,5750],{"style":947},[572,5748],{"className":5749,"style":951},[651],[572,5751,915],{"className":5752},[623,663],[572,5754,5755,5758],{"style":947},[572,5756],{"className":5757,"style":951},[651],[572,5759,5761],{"className":5760,"style":964},[963],[572,5762,5764],{"className":5763,"style":969},[968],[971,5765,5766],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,5767],{"d":982},[572,5769,687],{"className":5770},[761],[572,5772],{"className":5773,"style":766},[765],[572,5775,5777],{"className":5776},[623,935],[572,5778,5780],{"className":5779},[635],[572,5781,5783],{"className":5782},[639],[572,5784,5786,5794],{"className":5785,"style":931},[643],[572,5787,5788,5791],{"style":947},[572,5789],{"className":5790,"style":951},[651],[572,5792,1000],{"className":5793,"style":770},[623,663],[572,5795,5796,5799],{"style":947},[572,5797],{"className":5798,"style":951},[651],[572,5800,5802],{"className":5801,"style":964},[963],[572,5803,5805],{"className":5804,"style":969},[968],[971,5806,5807],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,5808],{"d":982},[572,5810,693],{"className":5811},[774],[572,5813],{"className":5814,"style":778},[765],[572,5816,1175],{"className":5817},[782],[572,5819],{"className":5820,"style":778},[765],[572,5822,5824,5827,5830,5865,5868,5871,5874,5909],{"className":5823},[614],[572,5825],{"className":5826,"style":749},[618],[572,5828,2030],{"className":5829},[753],[572,5831,5833],{"className":5832},[623,935],[572,5834,5836],{"className":5835},[635],[572,5837,5839],{"className":5838},[639],[572,5840,5842,5850],{"className":5841,"style":931},[643],[572,5843,5844,5847],{"style":947},[572,5845],{"className":5846,"style":951},[651],[572,5848,915],{"className":5849},[623,663],[572,5851,5852,5855],{"style":947},[572,5853],{"className":5854,"style":951},[651],[572,5856,5858],{"className":5857,"style":964},[963],[572,5859,5861],{"className":5860,"style":969},[968],[971,5862,5863],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,5864],{"d":982},[572,5866,2030],{"className":5867},[774],[572,5869],{"className":5870,"style":766},[765],[572,5872,2030],{"className":5873},[753],[572,5875,5877],{"className":5876},[623,935],[572,5878,5880],{"className":5879},[635],[572,5881,5883],{"className":5882},[639],[572,5884,5886,5894],{"className":5885,"style":931},[643],[572,5887,5888,5891],{"style":947},[572,5889],{"className":5890,"style":951},[651],[572,5892,1000],{"className":5893,"style":770},[623,663],[572,5895,5896,5899],{"style":947},[572,5897],{"className":5898,"style":951},[651],[572,5900,5902],{"className":5901,"style":964},[963],[572,5903,5905],{"className":5904,"style":969},[968],[971,5906,5907],{"xmlns":973,"width":974,"height":975,"style":976,"viewBox":977,"preserveAspectRatio":978},[980,5908],{"d":982},[572,5910,2030],{"className":5911},[774]," zeigen. Das ist schwächer.",[1768,5914,5915,5916,891],{},"Normen quadrieren und vergessen, auch das Skalarprodukt zu quadrieren: Richtig ist ",[572,5917,5919,5985],{"className":5918},[575],[572,5920,5922],{"className":5921},[579],[581,5923,5924],{"xmlns":583},[585,5925,5926,5982],{},[588,5927,5928,5930,5936,5938,5944,5950,5952,5954,5960,5966,5968,5970,5976],{},[678,5929,681],{"stretchy":680},[911,5931,5932,5934],{"accent":610},[594,5933,915],{},[678,5935,918],{},[678,5937,687],{"separator":610},[911,5939,5940,5942],{"accent":610},[594,5941,1000],{},[678,5943,918],{},[591,5945,5946,5948],{},[678,5947,693],{"stretchy":680},[1150,5949,1172],{},[678,5951,1175],{},[678,5953,2030],{"stretchy":680},[911,5955,5956,5958],{"accent":610},[594,5957,915],{},[678,5959,918],{},[591,5961,5962,5964],{},[678,5963,2030],{"stretchy":680},[1150,5965,1172],{},[729,5967,731],{},[678,5969,2030],{"stretchy":680},[911,5971,5972,5974],{"accent":610},[594,5973,1000],{},[678,5975,918],{},[591,5977,5978,5980],{},[678,5979,2030],{"stretchy":680},[1150,5981,1172],{},[602,5983,5984],{"encoding":604},"\\langle \\vec u, \\vec v \\rangle^2 \\le \\lVert \\vec u \\rVert^2\\,\\lVert \\vec v 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für orthogonale Vektoren erwarten. Dort ist die linke Seite ",[572,6258,6260,6273],{"className":6259},[575],[572,6261,6263],{"className":6262},[579],[581,6264,6265],{"xmlns":583},[585,6266,6267,6271],{},[588,6268,6269],{},[1150,6270,1804],{},[602,6272,1804],{"encoding":604},[572,6274,6276],{"className":6275,"ariaHidden":610},[609],[572,6277,6279,6282],{"className":6278},[614],[572,6280],{"className":6281,"style":2363},[618],[572,6283,1804],{"className":6284},[623],", Gleichheit gilt nur bei parallelen Vektoren (oder wenn ein Vektor ",[572,6287,6289,6307],{"className":6288},[575],[572,6290,6292],{"className":6291},[579],[581,6293,6294],{"xmlns":583},[585,6295,6296,6304],{},[588,6297,6298],{},[911,6299,6300,6302],{"accent":610},[1150,6301,1804],{},[678,6303,918],{},[602,6305,6306],{"encoding":604},"\\vec 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ist).",{"title":6353,"searchDepth":6354,"depth":6354,"links":6355},"",2,[6356,6357,6358],{"id":561,"depth":6354,"text":562},{"id":3275,"depth":6354,"text":3276},{"id":5657,"depth":6354,"text":5658},[6360,6363,6366,6368,6370,6372,6375,6378,6381],{"lhs":6361,"rhs":6362},"(a1^2 + a2^2 + a3^2)*(b1^2 + b2^2 + b3^2) - (a1*b1 + a2*b2 + a3*b3)^2","(a1*b2 - a2*b1)^2 + (a1*b3 - a3*b1)^2 + (a2*b3 - a3*b2)^2",{"lhs":6364,"rhs":6365},"abs(dot([a1, a2, a3], t*[a1, a2, a3]))","norm([a1, a2, a3]) * norm(t*[a1, a2, a3])",{"lhs":6367,"rhs":3596},"dot([1, 2, 2], [3, 0, 4])",{"lhs":6369,"rhs":3437},"norm([1, 2, 2])",{"lhs":6371,"rhs":3887},"norm([3, 0, 4])",{"lhs":6373,"rhs":6374},"round(acos(11\u002F15)*180\u002Fpi, 1)","42.8",{"lhs":6376,"rhs":6377},"3*(x^2 + y^2 + z^2) - (x + y + z)^2","(x - y)^2 + (x - z)^2 + (y - z)^2",{"lhs":6379,"rhs":6380},"(1\u002F2)^2","1\u002F4",{"lhs":6382,"rhs":6383},"1\u002F3 * 1","1\u002F3","\\vec u, \\vec v \\in V,\\ V \\text{ Vektorraum mit Skalarprodukt}","\u003Cspan 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15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.714em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2077em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>V\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">V\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">Vektorraum mit Skalarprodukt\u003C\u002Fspan>",null,"md","cauchy-schwarz","|\\langle \\vec u, \\vec v \\rangle| \\le \\lVert \\vec u \\rVert \\cdot \\lVert \\vec v \\rVert","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmo stretchy=\"false\">⟨\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>u\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>v\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">⟩\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmo>≤\u003C\u002Fmo>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>u\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>v\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">|\\langle \\vec u, \\vec v \\rangle| \\le \\lVert \\vec u \\rVert \\cdot \\lVert \\vec v \\rVert\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mopen\">⟨\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.714em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2077em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.714em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2077em;\">\u003Cspan class=\"katex-overlay\" 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height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">∥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" 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class=\"mclose\">∥\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fcauchy-schwarz-ungleichung",[],[6397,6398,6399,6400],"dot-product","angle-between-vectors","vector-length","triangle-inequality",{"title":548,"description":6353},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fcauchy-schwarz-ungleichung","Der Betrag des Skalarprodukts zweier Vektoren ist höchstens das Produkt ihrer Längen; Gleichheit nur bei parallelen Vektoren.",[6405,6406,6407,6408],"ungleichung","skalarprodukt","norm","abschätzung",[6410,6414,6418],{"symbol":6411,"meaning":6412,"symbolHtml":6413},"\\vec u, \\vec v","Vektoren eines Raums mit Skalarprodukt, z.B. des Rⁿ","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover 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style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2077em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">⟩\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":6419,"meaning":6420,"symbolHtml":6421},"\\lVert \\vec u \\rVert","Norm (Länge) von u, gleich der Wurzel aus ⟨u, u⟩","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>u\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">∥\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lVert \\vec u \\rVert\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">∥\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.714em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2077em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">∥\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","dZ6gFTjYTh-0t27Ez7f6KVunhHUCFbNztCJ4RYsmLnM",[6424,6426,6429,6432],{"stem":6425,"title":2376,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fdreiecksungleichung",{"stem":6427,"title":6428,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fbetrag-eines-vektors","Betrag eines Vektors",{"stem":6430,"title":6431,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fskalarprodukt","Skalarprodukt",{"stem":6433,"title":6434,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fwinkel-zwischen-zwei-vektoren","Winkel zwischen zwei Vektoren",1793824277391]