find a vector of magnitude 18 which is perpendicular to both the vectors 4i-j+3kand -2i+j-2k

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Solution

Dear Student,

$\overset{⇀}{a} = 4 \hat{i} - \hat{j} + 3 \hat{k} \overset{⇀}{b} = - 2 \hat{i} + \hat{j} - 2 \hat{k} f i n d a \times b t h e v e c t o r t h a t c o m e s w i l l b e p e r p e n d i c u l a r t o b o t h a a n d b \overset{⇀}{a} \times \overset{⇀}{b} = |\begin{matrix} i & j & k \\ 4 & - 1 & 3 \\ - 2 & 1 & - 2 \end{matrix}| = i (2 - 3) - j (- 8 + 6) + k (4 - 2) = - \hat{i} + 2 \hat{j} + 2 \hat{k} u n i t v e c t o r i n t h i s d i r e c t i o n = \frac{- \hat{i} + 2 \hat{j} + 2 \hat{k}}{\sqrt{1 + 4 + 4}} = \frac{- \hat{i} + 2 \hat{j} + 2 \hat{k}}{3} v e c t o r o f m a g n i t u d e 18 a n d p e r p e n d i c u l a r t o b o t h t h e v e c t o r = 18 (\frac{- \hat{i} + 2 \hat{j} + 2 \hat{k}}{3}) = - 6 \hat{i} + 12 \hat{j} + 12 \hat{k}$

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