First slide

Direction ratios and direction cosines

Question

$A = (- 1, 2, - 3), B = (5, 0, - 6), C = (0, 4, - 1)$ . If $l, m, n$ be the direction cosines of the internal angular bisector of $∠ B A C$ , then $l - m - n =$

Moderate

A

$\frac{28}{\sqrt{714}}$
B

$\frac{22}{\sqrt{714}}$
C

$\frac{38}{\sqrt{714}}$
D

$\frac{12}{\sqrt{714}}$

Solution

The vector along the internal angular bisector of $\bar{a}, \bar{b}$ is $(\frac{\bar{a}}{|a|} \pm \frac{\bar{b}}{|b|})$
Here $\bar{a} = \bar{A B} and \bar{b} = \bar{A C}$
It implies that $\bar{a} = 6 i - 2 j - 3 k and \bar{b} = i + 2 j + 2 k$
Hence, the vector along the internal angular bisector of $∠ B A C$ is $(\frac{\bar{a}}{|a|} + \frac{\bar{b}}{|b|}) = \frac{25 i + 8 j + 5 k}{21}$
Therefore, the direction cosines are $〈\frac{25}{\sqrt{714}}, \frac{8}{\sqrt{714}}, \frac{5}{\sqrt{714}}〉$
It implies that $l - m - n = \frac{12}{\sqrt{714}}$

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