Let u- and v- be two unit vectors- If w- is a vector such that w-w-u-v- then u-v-w- is equal toA- 1-v-w-B- 1- w -2C- - w -2- u - w 2D- - w -2- u - v 2

We have, nbsp;w+(w×u)=v ...(1) Taking dot product with v, we get (w+(w×u))·v=v·v ⇒w·v+[uvw]=1 (∵ |v|=1) ⇒ [uvw]=1 v·w ...(2) Now, taking cross pr ...

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Byju's Answer

Standard XII

Mathematics

Section Formula

Let u⃗ and v⃗...

Question

Let $\to u$ and $\to v$ be two unit vectors. If $\to w$ is a vector such that $\to w + (\to w \times \to u) = \to v$ , then $\to u \cdot (\to v \times \to w)$ is equal to

1 - \to v \cdot \to w

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1 - | \to w |^{2}

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| \to w |^{2} - (\to u \cdot \to w)^{2}

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| \to w |^{2} - (\to u \times \to v)^{2}

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Solution

The correct options are
A $1 - \to v \cdot \to w$
B $1 - | \to w |^{2}$
C $| \to w |^{2} - (\to u \cdot \to w)^{2}$
We have, $\to w + (\to w \times \to u) = \to v . . . (1)$
Taking dot product with $\to v$ , we get
$(\to w + (\to w \times \to u)) \cdot \to v = \to v \cdot \to v$
$\Rightarrow \to w \cdot \to v + [\to u \to v \to w] = 1 (∵ | \to v | = 1)$
$\Rightarrow [\to u \to v \to w] = 1 - \to v \cdot \to w . . . (2)$

Now, taking cross product of eqn $(1)$ with $\to u$ , we get
$\to u \times \to w + \to u \times (\to w \times \to u) = \to u \times \to v$
$\Rightarrow \to u \times \to w + (\to u \cdot \to u) \to w - (\to u \cdot \to w) \to u = \to u \times \to v$
$\Rightarrow \to u \times \to w + \to w - (\to u \cdot \to w) \to u = \to u \times \to v (∵ | \to u | = 1)$
Taking dot product with $\to w$ , we get
$\to w \cdot (\to u \times \to w) + \to w \cdot \to w - (\to u \cdot \to w) (\to w \cdot \to u) = \to w \cdot (\to u \times \to v)$
$\Rightarrow | \to w |^{2} - (\to u \cdot \to w)^{2} = [\to u \to v \to w]$

Now, taking dot product of eqn $(1)$ with $\to w$ , we get
$\to w \cdot \to w + \to w \cdot (\to w \times \to u) = \to w \cdot \to v$
$\Rightarrow | \to w |^{2} + 0 = \to v \cdot \to w$
$\Rightarrow | \to w |^{2} = \to v \cdot \to w$

Substituting the value of $\to v \cdot \to w$ in eqn $(2)$ , we get
$[\to u \to v \to w] = 1 - | \to w |^{2}$

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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

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is equal to that of

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and vectors

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

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\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

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\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. Let

\to u

\to v

\to w

be such that

| \to u |

= 1,

| \to v |

= 2,

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= 3 . If the projection

\to v

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and

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

equals

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Section Formula

Standard XII Mathematics

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Let →u and →v be two unit vectors. If →w is a vector such that →w+(→w×→u)=→v, then →u⋅(→v×→w) is equal to

Let $\to u$ and $\to v$ be two unit vectors. If $\to w$ is a vector such that $\to w + (\to w \times \to u) = \to v$ , then $\to u \cdot (\to v \times \to w)$ is equal to