First slide

Direction ratios and direction cosines

Question

The angle between two straight lines is $α$ . One line has direction cosines $(\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2})$ and the other line has direction ratios $(0, 1, 2)$ . Then $\tan α =$

Moderate

A

$\frac{3}{2 \sqrt{5}}$
B

$\frac{\sqrt{11}}{2 \sqrt{5}}$
C

$\frac{\sqrt{11}}{3}$
D

$\frac{3}{4 \sqrt{11}}$

Solution

If $θ$ is the acute angle between two lines having direction ratios $〈a_{1}, b_{1}, c_{1}〉$ and $〈a_{2}, b_{2}, c_{2}〉$
then $\cos θ = |\frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}} \cdot \sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}}|$
Given $α$ is angle between two lines having direction ratios $(\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2})$ and $(0, 1, 2)$
Hence,
$\begin{matrix} \cos α = \frac{\frac{1}{\sqrt{2}} (0) + \frac{1}{2} (1) + \frac{1}{2} (2)}{1 \cdot \sqrt{0^{2} + 1^{2} + 2^{2}}} \\ = \frac{3}{2 \sqrt{5}} \end{matrix}$
Use right angled triangle, to get the value of $\tan α$
Therefore, $\tan α = \frac{\sqrt{11}}{3}$

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