First slide

Introduction to 3-D Geometry

Question

The direction cosines of a line satisfy the relations $λ (l + m) = n$ and $m n + n l + l m = 0$ . The value of $λ$ , for which the two lines are perpendicular to each other, is

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

$\frac{1}{4}$

Solution

Eliminating n, we get $λ {(l + m)}^{2} + l m = 0$
$\Rightarrow \frac{λ l^{2}}{m^{2}} + (2 λ + 1) \frac{l}{m} + λ = 0$
$∴ \frac{l_{1} l_{2}}{m_{1} m_{2}} = 1$ [product of roots $\frac{l_{1}}{m_{1}} & \frac{l_{2}}{m_{2}}$ ]
Further eliminating m, we get $λ l^{2} - \ln - n^{2} = 0$
$\Rightarrow \frac{l_{1} l_{2}}{n_{1} n_{2}} = \frac{1}{λ}$ [product of roots $\frac{l_{1}}{n_{1}} & \frac{l_{2}}{n_{2}}$ ]
give the lines with DC's $(l_{1}, m_{1}, n_{1})$ and $(l_{2}, m_{2}, n_{2})$ are perpendicular
we have $l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$
$\Rightarrow 1 + 1 - λ = 0 \Rightarrow λ = 2$

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