First slide

Direction ratios and direction cosines

Question

If the direction cosines $(l, m, n)$ of two perpendicular lines are connected by the relations $λ l + 3 m - 5 n = 0 and 10 l^{2} + m^{2} - 5 n^{2} = 0$ then $\frac{λ}{\sqrt{5}}$ is

Moderate

A

$\pm 3$
B

$\pm 1$
C

$\pm 4$
D

$\pm 5$

Solution

The direction cosines of two lines are related by relations $a l + b m + c n = 0$ and $u l^{2} + v m^{2} + w n^{2} = 0$ . If these two lines are perpendicular then the condition is $\sum a^{2} (v + w) = 0$
Here $a = λ, b = 3, c = - 5, u = 10, v = 1, w = - 5$
Hence,
$\begin{matrix} \sum a^{2} (v + w) = 0 \\ λ^{2} (- 4) + 9 (5) + 25 (11) = 0 \\ - 4 λ^{2} + 320 = 0 \end{matrix}$
Therefore, $λ = \pm 4 \sqrt{5} \Rightarrow \frac{λ}{\sqrt{5}} = \pm 4$

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