For any two vectors u- v- find u-v-2-u-v-2 - -u-2-v-2-

| U . V #160; | =| U #160; | .| V #160; | cosθ #160; | U × V #160; | =| U #160; | .| V #160; | sinθ #160; where θ ...

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Byju's Answer

Standard XII

Physics

Vector Component

For any two v...

Question

For any two vectors $\to u$ & $\to v$ find $(\to u . \to v)^{2} + | \to u \times \to v |^{2} - | \to u |^{2} | \to v |^{2}$ .

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Solution

$∣ ∣ ¯ ¯¯ ¯ U . ¯ ¯¯ ¯ V ∣ ∣ = ∣ ∣ ¯ ¯¯ ¯ U ∣ ∣ . ∣ ∣ ¯ ¯¯ ¯ V ∣ ∣ cos θ$

$∣ ∣ ¯ ¯¯ ¯ U \times ¯ ¯¯ ¯ V ∣ ∣ = ∣ ∣ ¯ ¯¯ ¯ U ∣ ∣ . ∣ ∣ ¯ ¯¯ ¯ V ∣ ∣ sin θ$

$w h e r e θ i s t h e a n g l e b e t w e e n ¯ ¯¯ ¯ U a n d ¯ ¯¯ ¯ V .$

$⟹ {∣ ∣ ¯ ¯¯ ¯ U . ¯ ¯¯ ¯ V ∣ ∣}^{2} + {∣ ∣ ¯ ¯¯ ¯ U \times ¯ ¯¯ ¯ V ∣ ∣}^{2}$

$= {∣ ∣ ¯ ¯¯ ¯ U ∣ ∣}^{2} {∣ ∣ ¯ ¯¯ ¯ V ∣ ∣}^{2} (cos^{2} θ + sin^{2} θ)$

$= {∣ ∣ ¯ ¯¯ ¯ U ∣ ∣}^{2} {∣ ∣ ¯ ¯¯ ¯ V ∣ ∣}^{2}$

$∴ {∣ ∣ ¯ ¯¯ ¯ U . ¯ ¯¯ ¯ V ∣ ∣}^{2} + {∣ ∣ ¯ ¯¯ ¯ U \times ¯ ¯¯ ¯ V ∣ ∣}^{2} - {∣ ∣ ¯ ¯¯ ¯ U ∣ ∣}^{2} . {∣ ∣ ¯ ¯¯ ¯ V ∣ ∣}^{2}$

$= {∣ ∣ ¯ ¯¯ ¯ U ∣ ∣}^{2} . {∣ ∣ ¯ ¯¯ ¯ V ∣ ∣}^{2} - {∣ ∣ ¯ ¯¯ ¯ U ∣ ∣}^{2} . {∣ ∣ ¯ ¯¯ ¯ V ∣ ∣}^{2} = 0$

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Similar questions

\to u, \to v, \to w

be such that

| \to u | = 1, | \to v | = 2, | \to w | = 3.

If the projection of

\to v

along

\to u

is equal to that of

\to w

along

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and vectors

\to v, \to w

are perpendicular to each other then

| \to u - \to v + \to w |

equals

Q. Let

\to u, \to v, \to w

be such that

| \to u | = 1, | \to v | = 2, | \to w | = 3

. If the projection

\to v

along

\to u

is equal to that of

\to w

along

\to u

and

\to v, \to w

are perpendicular to each other then

| \to u - \to v + \to w |

equals:

Q. Let

\to u, \to v, \to w

be such that

| \to u | = 1, | \to v | = 2, | \to w | = 3

. If the projection of

\to v

along

\to u

is equal to that of

\to w

along

\to u

and

\to v, \to w

are perpendicular to each other, then

| \to u - \to v + \to w |

equals

Q. Let

\to u, \to v

and

\to w

be vectors such that

\to u + \to v + \to w = \to 0 .

| \to u | = 3, | \to v | = 4

and

| \to w | = 5,

then

\to u . \to v + \to v . \to w + \to w . \to u

Q. If in a triangle

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\to B C = \frac{\to u}{| \to u |} - \frac{\to v}{| \to v |}

and

\to A C = \frac{2 \to u}{| \to u |}

, where

| \to u | \neq | \to v |, t h e n

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For any two vectors →u & →v find (→u.→v)2+|→u×→v|2−|→u|2|→v|2.

For any two vectors $\to u$ & $\to v$ find $(\to u . \to v)^{2} + | \to u \times \to v |^{2} - | \to u |^{2} | \to v |^{2}$ .