Find the unit vectors perpendicular to the following 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{35}} (3 i + j - 5 k)

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\frac{1}{\sqrt{27}} (i - j - 5 k)

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\frac{1}{\sqrt{27}} (i - j + 5 k)

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Solution

The correct option is C $\frac{1}{\sqrt{35}} (3 i - j - 5 k)$
$\to a = 2^i +^j +^k$

$\to b =^i -^2 j +^k$

Vector perpendicular to Given vectors $\to a, \to b$

$\Rightarrow \to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & 1 1 & - 2 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$\Rightarrow \to a \times \to b =^i (1 - (- 2)) -^j (2 - 1) +^k (- 4 - 1)$

$\Rightarrow \to a \times \to b = 3^i -^j -^5 k$

Unit vector required is $\frac{^3 i -^j - 5^k}{∣ ∣ 3^i -^j - 5^k ∣ ∣}$ $= \frac{1}{\sqrt{35}} (3^i -^j -^5 k)$

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