First slide

Algebra of vectors

Question

$\vec{a} = - 4 \hat{i} + 3 \hat{k}; \vec{b} = 14 \hat{i} + 2 \hat{j} - 5 \hat{k}$ vectors are laid off from one point. Vector $\vec{d}$ which is being laid off from the same point dividing the angle between vector, $\vec{a} and \vec{b}$ in equal halves and having the magnitude $\sqrt{6}$ , is

Moderate

A

$\hat{i} + \hat{j} + 2 \hat{k}$
B

$\hat{i} - \hat{j} + 2 \hat{k}$
C

$\hat{i} + \hat{j} - 2 \hat{k}$
D

$2 \hat{i} - \hat{j} - 2 \hat{k}$

Solution

$\hat{a} = \frac{- 4 i + 3 \hat{k}}{5}; \hat{b} = \frac{14 \hat{i} + 2 \hat{j} - 5 \hat{k}}{15}$
$A vector \vec{V} bisecting the angle between \vec{a} and \vec{b} is$
$\begin{array}{l} \vec{V} = \hat{a} + \hat{b} \\ = \frac{- 12 \hat{i} + 9 \hat{k} + 14 \hat{i} + 2 \hat{j} - 5 \hat{k}}{15} \\ = \frac{2 \hat{i} + 2 \hat{j} + 4 \hat{k}}{15} \end{array}$
$Required vector \vec{d} = \sqrt{6} \hat{V} = \hat{i} + \hat{j} + 2 \hat{k}$

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