If u-v-w are non-coplanar vectors and p-q are real numbers then the equality -3u pv pw-pv w qu-2w qv qu-0 holds for

Given : [3u pv pw] [pv w qu]+[2w qv qu]=0 ⇒ 3p^2[u v w] pq[u v w] 2q^2[u v w]=0 ⇒ nbsp;[u v w](3p^2 pq 2q^2)=0 ⇒ nbsp;3p^2 pq 2q^2=0; (∵ [u v w]0) ⇒ (p q)(3 ...

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Standard XII

Mathematics

Dot Product of Two Vectors

If u,v,w are ...

Question

If $\to u, \to v, \to w$ are non-coplanar vectors and $p, q$ are real numbers then the equality $[3 \to u p \to v p \to w] - [p \to v \to w q \to u] + [2 \to w q \to v q \to u] = 0$ holds for

exactly two values of

(p, q)

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more than two but not all values of

(p, q)

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all values of

(p, q)

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exactly one value of

(p, q)

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Solution

The correct option is B more than two but not all values of $(p, q)$
Given :
$[3 \to u p \to v p \to w] - [p \to v \to w q \to u] + [2 \to w q \to v q \to u] = 0 \Rightarrow 3 p^{2} [\to u \to v \to w] - p q [\to u \to v \to w] - 2 q^{2} [\to u \to v \to w] = 0 \Rightarrow [\to u \to v \to w] (3 p^{2} - p q - 2 q^{2}) = 0 \Rightarrow 3 p^{2} - p q - 2 q^{2} = 0; (∵ [\to u \to v \to w] \neq 0) \Rightarrow (p - q) (3 p + 2 q) = 0$
$\Rightarrow p = q$ or $p = - \frac{2}{3} q$
so, there are more than two values of $(p, q)$ but not all values of $(p, q)$

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Similar questions

Q. If

\to u, \to v, \to w

are non-coplanar vectors and

p, q

are real numbers, then the equality

[\begin{matrix} 3 \to u & p \to v & p \to w \end{matrix}] - [\begin{matrix} p \to v & q \to w & \to u \end{matrix}] - [\begin{matrix} 2 \to w & q \to v & q \to u \end{matrix}] = 0

holds for

Q. If

\to u, \to v

and

\to w

are three non-coplanar vectors, then

(\to u + \to v - \to w) . (\to u - \to v) \times (\to v - \to w)

equals

Q. If

\to u, \to v, \to w

are three non-zero and non-coplanar vectors, then

(\to u + \to v - \to w) \cdot [(\to u - \to v) \times (\to v - \to w)]

is equal to:

Q. If

\to u, \to v, \to w

are three noncoplanar vectors then

[\to u + \to v - \to w \to u - \to v \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

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Dot Product of Two Vectors

Standard XII Mathematics

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If →u,→v,→w are non-coplanar vectors and p,q are real numbers then the equality [3→u p→v p→w]−[p→v →w q→u]+[2→w q→v q→u]=0 holds for

If $\to u, \to v, \to w$ are non-coplanar vectors and $p, q$ are real numbers then the equality $[3 \to u p \to v p \to w] - [p \to v \to w q \to u] + [2 \to w q \to v q \to u] = 0$ holds for