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Multiplication with Vectors

Find the vect...

Question

Find the vectors of magnitude $6$ which are perpendicular to both the vectors $4 \to i - \to j + 3 \to k$ and $- 2 \to i + \to j - 2 \to k$ .

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Solution

Let $\to a = 4 \to i - \to j + 3 \to k$
and $\to b = - 2 \to i + \to j - 2 \to k$
Therefore, $\to a \times \to b = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to i & \to j & \to k 4 & - 1 & 3 - 2 & 1 & - 2 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= \to i (2 - 3) - \to j (- 8 + 6) + \to k (4 - 2)$
$= - \to i + 2 \to j + 2 \to k$
$\Rightarrow ∣ ∣ \begin{matrix} \to a \times \to b \end{matrix} ∣ ∣ = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$
The required vector $= \frac{\pm 6 (\to a \times \to b)}{∣ ∣ \begin{matrix} \to a \times \to b \end{matrix} ∣ ∣}$
$= \frac{\pm 6 (- \to i + 2 \to j + 2 \to k)}{3}$
$= \pm (- 2 \to i + 4 \to j + 4 \to k)$

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