If $\to a =^i + 2^j +^k, \to b = 2^i +^j$ and $\to c = 3^i - 4^j - 5^k$ , then find a unit vector perpendicular to both of the vectors $(\to a - \to b)$ and $(\to c - \to b)$ .

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Solution

We have $\to a - \to b = -^i +^j +^k \to c - \to b =^i - 5^j - 5^k$
$(\to a - \to b) \times (\to c - \to b) = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k - 1 & 1 & 1 1 & - 5 & - 5 \end{matrix} ∣ ∣ ∣ ∣ ∣ = - 4^j + 4^k$
The required unit vector is: $\frac{(\to a - \to b) \times (\to c - \to b)}{| (\to a - \to b) \times (\to c - \to b) |} = \frac{- 4^j + 4^k}{4 \sqrt{2}} = \frac{-^j +^k}{\sqrt{2}} .$

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