First slide

Direction ratios and direction cosines

Question

The acute angle between the two lines whose direction ratios are connected by $l + m - n = 0 a n d l^{2} + m^{2} - n^{2} = 0$ is

Moderate

A

$\frac{π}{3}$
B

0
C

$\frac{π}{6}$
D

$\frac{π}{4}$

Solution

Suppose that $n = 1$
Hence the equations become $l + m = 1$ and $l^{2} + m^{2} = 1$
Substitute $m = 1 - l$ in the equation $l^{2} + m^{2} = 1$ and then solve for $l$
It implies that $l = 1, 0$ and $m = 0, 1$
Therefore, the direction ratios are $〈1, 0, 1〉, 〈0, 1, 1〉$
If $θ$ is the acute angle between two lines having direction ratios $〈a_{1}, b_{1}, c_{1}〉$ and $〈a_{2}, b_{2}, c_{2}〉$
then $\cos θ = |\frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}} \cdot \sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}}|$
Suppose that $θ$ be the angle between the lines whose direction ratios are $〈1, 0, 1〉, 〈0, 1, 1〉$
Then $\cos θ = \frac{1}{\sqrt{2} \sqrt{2}} = \frac{1}{2}$
Therefore, the measure of angle $θ$ is $\frac{π}{3}$

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