Let a = i - 2 j + 3 k , b = 2 i + j - k , c = i + j + k and v = a × b . I f v · c = 11 and the length of the projection of b on c, is p, then 9 p 2 is equal to. [2026]

Question

Let $\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}, \vec{c} = λ \hat{i} + \hat{j} + \hat{k}$ and $\vec{v} = \vec{a} \times \vec{b} . If \vec{v} \cdot \vec{c} = 11$ and the length of the projection of $\vec{b} on \vec{c}$ , is p, then $9 p^{2}$ is equal to. [2026]

Smart Achievers · Accepted Answer

(4)

$\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k},$ $\vec{b} = 2 \hat{i} + \hat{j} - \hat{k},$ $\vec{c} = λ \hat{i} + \hat{j} + \hat{k}, and \vec{v} = \vec{a} \times \vec{b} .$ $If \vec{v} \cdot \vec{c} = 11$

$\vec{v} = (\vec{a} \times \vec{b}) = (- \hat{i} + 7 \hat{j} + 5 \hat{k})$

$\vec{v} \cdot \vec{c} = 11 = (- \hat{i} + 7 \hat{j} + 5 \hat{k}) \cdot (λ \hat{i} + \hat{j} + \hat{k}) = 11$

$\Rightarrow - λ + 7 + 5 = 11$

$\Rightarrow λ = 1$

$Length of projection of \vec{b} on \vec{c} = \vec{b} \cdot \hat{c}$

$\Rightarrow$ $| (2 \hat{i} + \hat{j} - \hat{k}) \cdot \frac{(\hat{i} + \hat{j} + \hat{k})}{\sqrt{3}} |$ $= \frac{2 + 1 - 1}{\sqrt{3}} = p = \frac{2}{\sqrt{3}}$

$\Rightarrow 9 p^{2} = 9 (\frac{4}{3}) = 12$

Solution

1 4

2 9

3 6

4 12

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Solution

Q. Let a→=i^-2j^+3k^, b→=2i^+j^-k^, c→=λi^+j^+k^ and v→=a→×b→. If v→·c→=11 and the length of the projection of b→ on c→, is p, then 9p2 is equal to. [2026]

1 4

2 9

3 6

4 12

Ans. (4) a→=i^-2j^+3k^, b→=2i^+j^-k^, c→=λi^+j^+k^, and v→=a→×b→. If v→·c→=11 v→=(a→×b→)=(-i^+7j^+5k^) v→·c→=11=(-i^+7j^+5k^)·(λi^+j^+k^)=11 ⇒-λ+7+5=11 ⇒λ=1 Length of projection of b→ on c→=b→·c^ ⇒|(2i^+j^-k^)·(i^+j^+k^)3|=2+1-13=p=23 ⇒9p2=9(43)=12

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