Find the unit vectors perpendicular to given pair of vectors: $2 i + j + k, i - 2 j + k$

\frac{1}{\sqrt{35}} (3 i + j - 5 k) .

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\frac{1}{\sqrt{11}} (i - j - 3 k) .

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\frac{1}{\sqrt{35}} (3 i - j - 5 k) .

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\frac{1}{\sqrt{11}} (i + j + 3 k) .

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Solution

The correct option is D $\frac{1}{\sqrt{35}} (3 i - j - 5 k) .$
Let $\to a, \to b$ be two non-zero vectors.
Given $2 i + j + k, i - 2 j + k$
$(2 i + j + k) \times (i - 2 j + k) = - 4 k - 2 j - k + i + j + 2 i = 3 i - j - 5 k$
$| (2 i + j + k) \times (i - 2 j + k) | = | 3 i - j - 5 k | = \sqrt{35}$

$\frac{(2 i + j + k) \times (i - 2 j + k)}{| (2 i + j + k) \times (i - 2 j + k) |} = \frac{3 i - j - 5 k}{\sqrt{35}}$

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\vec{a} = \hat{i} - \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}, \vec{c} = λ \hat{i} - \hat{j} + λ \hat{k}

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A unit vector coplanar with $\to i + \to j + 3 \to k and \to i + 3 \to j + \to k$ and perpendicular to $\to i + \to j + \to k$ is

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