First slide

Algebra of vectors

Question

$A vector, \vec{c} directed along the internal bi-sector of the angle between the vectors \vec{a} = 7 \hat{i} - 4 \hat{j} - 4 \hat{k} and \vec{b} = - 2 \hat{i} - \hat{j} + 2 \hat{k}, with | \vec{c} | = 5 \sqrt{6}, is$

Moderate

A

$\frac{5}{3} (i - 7 j + 2 k)$
B

$\frac{5}{3} (5 i + 5 j + 2 k)$
C

$\frac{5}{3} (i + 7 j + 2 k)$
D

$\frac{5}{3} (- 5 i + 5 j + 2 k)$

Solution

$T h e r e q u i r e d v e c t o r s \vec{c} i s g i v e n b y$ $λ (\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|})$
$n o w \frac{\vec{a}}{|\vec{a}|} = \frac{1}{9} (7 \hat{i} - 4 \hat{j} - 4 \hat{k}) a n d \frac{\vec{b}}{|\vec{b}|} = \frac{1}{3} (- 2 \hat{i} - \hat{j} + 2 \hat{k})$
$\Rightarrow \vec{c} = λ (\frac{1}{9} \hat{i} - \frac{7}{9} \hat{j} + \frac{2}{9} \hat{k}) \Rightarrow {|\vec{c}|}^{2} = λ^{2} \times \frac{54}{81}$
$Therefore, λ^{2} = 225 or λ = \pm 15$
$\Rightarrow \vec{c} = \pm \frac{5}{3} (\hat{i} - 7 \hat{j} + 2 \hat{k})$

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