A unit vector in the plane of the vectors 2i¯+j¯+k¯, i¯−j¯+k¯ and orthogonal to 5i¯+2j¯+6k¯ is

(a¯×b¯)×c¯=(a¯⋅c¯)b¯−(b¯⋅c¯)a¯=18b¯−9a¯=9[(2,−2,2)−(2,1,1)]=9(0,−3,1)Unit vector =±110(3j¯−k¯)

A unit vector in the plane of the vectors 2i¯+j¯+k¯, i¯−j¯+k¯ and orthogonal to 5i¯+2j¯+6k¯ is

(a¯×b¯)×c¯=(a¯⋅c¯)b¯−(b¯⋅c¯)a¯=18b¯−9a¯=9[(2,−2,2)−(2,1,1)]=9(0,−3,1)Unit vector =±110(3j¯−k¯)

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A unit vector in the plane of the vectors $2 \bar{i} + \bar{j} + \bar{k}, \bar{i} - \bar{j} + \bar{k}$ and orthogonal to $5 \bar{i} + 2 \bar{j} + 6 \bar{k}$ is

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$\frac{2 \bar{i} + \bar{j} - 2 \bar{k}}{3}$

$\frac{6 \bar{i} - 6 \bar{k}}{\sqrt{61}}$

$\frac{3 \bar{j} - \bar{k}}{\sqrt{10}}$

$\frac{2 \bar{i} - 5 \bar{j}}{\sqrt{29}}$

answer is B.

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Detailed Solution

$\begin{array}{l} (\bar{a} \times \bar{b}) \times \bar{c} \\ = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{b} \cdot \bar{c}) \bar{a} \\ = 18 \bar{b} - 9 \bar{a} \\ = 9 [(2, - 2, 2) - (2, 1, 1)] \\ = 9 (0, - 3, 1) \end{array}$
Unit vector $= \pm \frac{1}{\sqrt{10}} (3 \bar{j} - \bar{k})$