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Addition of Vectors

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Question

write the number of vectors of unit length perpendicular to both the vectors $\to a = 2^i +^j + 2^k$ and $\to b =^j +^k$ .

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Solution

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

$\to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 2 & 1 & 2 0 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$=^i (1 - 2) -^j (2 - 0) +^k (2 - 0)$

$= \frac{-^i - 2^j + 2^k}{\sqrt{2^{2} + 2^{2} + 1}}$

$= \frac{-^i - 2^j + 2^k}{\sqrt{9}}$

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

$- (\to a \times \to b) = \frac{- (^i - 2^j + 2^k)}{3}$

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

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