a→=2i^-j^+k^, b→=i^+2j^-k^, c→=i^+j^-2k^. A vector coplanar with b→ and c→, whose projection on a→ is of magnitude 23 is

d=xi+yj+2k d b c=0 ⇒ xyz12-111-2=0 x(-3)-y(-1)+z(-1)=0 d·aa=23 ⇒ 2x-y+z6=23 ⇒ 2x-y+z=2verification of options

a→=2i^-j^+k^, b→=i^+2j^-k^, c→=i^+j^-2k^. A vector coplanar with b→ and c→, whose projection on a→ is of magnitude 23 is

d=xi+yj+2k d b c=0 ⇒ xyz12-111-2=0 x(-3)-y(-1)+z(-1)=0 d·aa=23 ⇒ 2x-y+z6=23 ⇒ 2x-y+z=2verification of options

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$\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{c} = \hat{i} + \hat{j} - 2 \hat{k} .$ A vector coplanar with $\vec{b}$ and $\vec{c}$ , whose projection on $\vec{a}$ is of magnitude $\sqrt{\frac{2}{3}}$ is

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$2 \hat{i} + 3 \hat{j} - 3 \hat{k}$

$- 2 \hat{i} - \hat{j} + \hat{k}$

$2 \hat{i} + 3 \hat{j} + 3 \hat{k}$

$2 \hat{i} + \hat{j} + 5 \hat{k}$

answer is A.

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Detailed Solution

$d = xi + yj + 2 k [d b c] = 0 \Rightarrow |\begin{matrix} x & y & z \\ 1 & 2 & - 1 \\ 1 & 1 & - 2 \end{matrix}| = 0 x (- 3) - y (- 1) + z (- 1) = 0 \frac{d \cdot a}{|a|} = \sqrt{\frac{2}{3}} \Rightarrow \frac{2 x - y + z}{\sqrt{6}} = \sqrt{\frac{2}{3}} \Rightarrow 2 x - y + z = 2$