First slide

Direction ratios and direction cosines

Question

The angle between two straight lines having direction cosines $〈\frac{l}{\sqrt{21}}, \frac{m}{\sqrt{21}}, \frac{1}{\sqrt{21}}〉 and 〈\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}}, \frac{3 l}{\sqrt{54}}〉$ is $90^{0}$ , then a pair of possible values of $l, m$ respectively are

Easy

A

2,4
B

-2,4
C

-2,-4
D

1,2

Solution

If the direction ratios of two perpendicular lines are proportional to $〈a_{1}, b_{1}, c_{1}〉, 〈a_{2}, b_{2}, c_{2}〉$ then $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Hence $\frac{l}{\sqrt{21}} \cdot \frac{3}{\sqrt{54}} + \frac{m}{\sqrt{21}} \cdot \frac{3}{\sqrt{54}} + \frac{1}{\sqrt{21}} \cdot \frac{3 l}{\sqrt{54}} = 0$
It implies that $6 l + 3 m = 0 \Rightarrow 2 l + m = 0$
Verify the options for suitable values of $l, m$

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