First slide

Introduction to 3-D Geometry

Question

If the angle between line with d.c’s $(- \frac{2}{\sqrt{21}}, \frac{a}{\sqrt{21}}, \frac{b}{\sqrt{21}})$ and other line with d.c’s $(\frac{3}{\sqrt{54}}, \frac{3}{\sqrt{54}}, - \frac{6}{\sqrt{54}})$ is 90⁰ then a pair of possible values of ‘a’ and ‘b’ respectively are

Moderate

A

-1, 4
B

4, 2
C

4, 1
D

-4, -2

Solution

${(\frac{- 2}{\sqrt{21}})}^{2} + {(\frac{a}{\sqrt{21}})}^{2} + {(\frac{b}{\sqrt{21}})}^{2} = 1$
$(∵ l^{2} + m^{2} + n^{2} = 1)$
$4 + a^{2} + b^{2} = 21$ $\Rightarrow a^{2} + b^{2} = 17....... (1)$
Angle between the given lines is 90⁰
$\Rightarrow (\frac{- 2}{\sqrt{21}}) (\frac{3}{\sqrt{54}}) + (\frac{a}{\sqrt{21}}) (\frac{3}{\sqrt{54}})$ $+ (\frac{b}{\sqrt{21}}) (\frac{- 6}{\sqrt{54}}) = 0$
$- 6 + 3 a - 6 b = 0$ $\Rightarrow 3 a - 6 b = 6$ $\Rightarrow a - 2 b = 2..... (2)$
Solving (1) & (2), we get a possible solution given by $a = 4 : b = 1$

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