If-u-and-v-are-vectors-in-R-3-then-prove-that-u-v-1-4-u-v-2-1-4-u-v-2-

Question Number 185695 by Spillover last updated on 25/Jan/23

I f \vec{u} a n d \vec{v} a r e v e c t o r s i n R^{3}

t h e n p r o v e t h a t

\vec{u} . \vec{v} = \frac{1}{4} ∥ \vec{u} + \vec{v} ∥^{2} - \frac{1}{4} ∥ \vec{u} - \vec{v} ∥^{2}

Answered by mahdipoor last updated on 26/Jan/23

u = (a, b, c) v = (x, y, z)

\frac{1}{4} ∣∣ u + v ∣ ∣^{2} - \frac{1}{4} ∣∣ u - v ∣ ∣^{2} =

\frac{1}{4} [{(a + x)}^{2} + {(b + y)}^{2} + {(c + z)}^{2}] -

\frac{1}{4} [{(a - x)}^{2} + {(b - y)}^{2} + {(c - z)}^{2}] =

\frac{1}{4} [4 a x + 4 b y + 4 c z] = a x + b y + c z = u . v

Answered by cortano1 last updated on 26/Jan/23

∣∣ \vec{u} + \vec{v} ∣ ∣^{2} = ∣∣ \vec{u} ∣ ∣^{2} + ∣∣ \vec{v} ∣ ∣^{2} + 2 \vec{u} . \vec{v}

∣∣ \vec{u} - \vec{v} ∣ ∣^{2} = ∣∣ \vec{u} ∣ ∣^{2} + ∣∣ \overset{⌣}{v} ∣ ∣^{2} - 2 \vec{u} . \vec{v}

∣ \vec{u} + \vec{v} ∣^{2} - ∣ \vec{u} - \vec{v} ∣^{2} = 4 \vec{u} . \vec{v}

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