The vector orthogonal to $¯ ¯ ¯ a = 2 ¯ i + ¯ j - 3 ¯ ¯ ¯ k, ¯ ¯ b = ¯ i - 2 ¯ j + ¯ ¯ ¯ k$ and having magnitude equal to $5$ units is

\pm \frac{5 \sqrt{3}}{3} (¯ i + ¯ j + ¯ ¯ ¯ k)

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\pm \frac{5 \sqrt{3}}{3} (¯ i + ¯ j - ¯ ¯ ¯ k)

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\pm \frac{5 \sqrt{3}}{3} (¯ i - ¯ j + ¯ ¯ ¯ k)

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\pm \frac{5 \sqrt{3}}{3} (- ¯ i + ¯ j + ¯ ¯ ¯ k)

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Solution

The correct option is B $\pm \frac{5 \sqrt{3}}{3} (¯ i + ¯ j + ¯ ¯ ¯ k)$
$\to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & - 3 1 & - 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 5^i - 5^j - 5^k$
$\to V = \to a \times \to b$
$| \to U | = | \to K | \to a \times \to b |$
$5 = | K | 5 \sqrt{3}$
$| K | = \frac{1}{\sqrt{3}}$
$K =_{-}^{+} \frac{1}{\sqrt{3}}$
$\to V =_{-}^{+} \frac{5}{\sqrt{3}} (^i +^j +^k)$

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