First slide

Introduction to 3-D Geometry

Question

If the dr’s of two lines are given by $3 l m - 4 l n + m n = 0$ and $l + 2 m + 3 n = 0$ then the angle between the lines is

Moderate

A

$\frac{π}{2}$
B

$\frac{π}{3}$
C

$\frac{π}{4}$
D

$\frac{π}{6}$

Solution

$l + 2 m + 3 n = 0$ $\Rightarrow l = - 2 m - 3 n$
$3 l m - 4 \ln + n m = 0$
$\Rightarrow 3 m (- 2 m - 3 n) - 4 n (- 2 m - 3 n) + m n = 0$
$\Rightarrow - 6 m^{2} - 9 m n + 8 m n + 12 n^{2} + m n = 0$
$\Rightarrow 12 n^{2} = 6 m^{2}$ $\Rightarrow m = \pm \sqrt{2} n, l = (+ 2 \sqrt{2}) n$
D.r’s of the lines are $(- 2 \sqrt{2} - 3, \sqrt{2}, 1) (2 \sqrt{2} - 3, - \sqrt{2}, 1)$
$∴ a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} =$ $(- 2 \sqrt{2} - 3) (2 \sqrt{2} - 3) - 2 + 1$ $= 9 - 8 - 2 + 1 = 0$
$\Rightarrow$ Required angle $= \frac{π}{2}$

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