First slide

Direction ratios and direction cosines

Question

If the direction ratios of two perpendicular straight lines are given by $(1, θ, θ^{3} - θ + 7) and (0, 4, θ) where θ \neq 0$ , then ${Tan}^{- 1} (\frac{θ - θ^{3}}{11}) =$

Easy

A

0
B

$π$
C

$\frac{π}{4}$
D

$\frac{π}{2}$

Solution

The lines having direction ratios $〈a_{1}, b_{1}, c_{1}〉$ and $〈a_{2}, b_{2}, c_{2}〉$ are perpendicular then $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Hence, $\begin{matrix} 4 θ + θ^{4} - θ^{2} + 7 θ = 0 \\ θ^{3} - θ + 11 = 0 \\ \frac{θ - θ^{3}}{11} = 1 \end{matrix}$
Therefore, ${Tan}^{- 1} (\frac{θ - θ^{3}}{11}) = \frac{π}{4}$

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