First slide

Algebra of vectors

Question

If $A (- 4, 0, 3) and B (14, 2, - 5)$ then which one of the following points lie on the bisector of the angle between $\vec{O A}$ and $\vec{O B}$ (O is the origin of reference)?

Moderate

A

(2,2, 4)
B

(2, 11, 5)
C

(-3, -3, -6)
D

(1,1, 2)

Solution

$\begin{array}{l} \vec{OA} = - 4 \hat{i} + 3 \hat{k}; \vec{OB} = 14 \hat{i} + 2 \hat{j} - 5 \hat{k} \\ \hat{a} = \frac{- 4 \hat{i} + 3 \hat{k}}{5}; \hat{b} = \frac{14 \hat{i} + 2 \hat{j} - 5 \hat{k}}{15} \\ \vec{r} = \frac{λ}{15} [- 12 \hat{i} + 9 \hat{j} + 14 \hat{i} + 2 \hat{j} - 5 \hat{k}] \\ = \frac{λ}{15} [2 \hat{i} + 2 \hat{j} + 4 \hat{k}] \\ = \frac{2 λ}{15} [\hat{i} + \hat{j} + 2 \hat{k}] \end{array}$

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