First slide

Direction ratios and direction cosines

Question

The direction ratios of two lines are given by $a + b + c = 0 and a b + b c - 2 c a = 0$ . Then the angle between the lines is

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{3}$
C

$\frac{π}{2}$
D

0

Solution

Suppose that $n = 1$
Hence, the equations become $a + b = 1$ and $a b - b + 2 a = 0$
Substitute $b = 1 - a$ in the equation $a b - b + 2 a = 0$ and then solve for $a$
It implies that $l = 2 \pm \sqrt{3}$ and $l = 2 \pm \sqrt{3}$
Therefore, the direction ratios are $〈2 + \sqrt{3}, - 1 - \sqrt{3}, - 1〉, 〈2 - \sqrt{3}, - 1 + \sqrt{3}, - 1〉$
Suppose that $θ$ be the angle between the lines whose direction ratios are $〈2 + \sqrt{3}, - 1 - \sqrt{3}, - 1〉, 〈2 - \sqrt{3}, - 1 + \sqrt{3}, - 1〉$
Hence, $\cos θ = \frac{4 - 3 + (1 - 3) + 1}{\sqrt{{(2 + \sqrt{3})}^{2} + {(- 1 - \sqrt{3})}^{2} + 1} \cdot \sqrt{{(2 - \sqrt{3})}^{2} + {(- 1 + \sqrt{3})}^{2} + 1}} = 0$
Therefore, the measure of angle $θ$ is $\frac{π}{2}$

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