If a line segment OP makes angles of π/4 and π/3 with X- axis and 7-axis, respectively Then, the direction cosines are

Correct Answer is : 12, 12, 12{1}{{2}},\ 1/2,\ {1}{{2}}21, 21, 21

Correct Answer is : 12, 12, 12{1}{{2}},\ 1/2,\ {1}{{2}}21, 21, 21

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VITEEE 2015 Paper

Section: Mathematics

Question : 98 of 120

Marks: +1, -0

\frac{\pi}{4}

\frac{\pi}{3}

Solution:

Let α, β and γ be the angles made by the line segment OP with X-axis, Y-axis and Z-axis, respectively.

Given , α =

\frac{\pi}{4}

and β =

\frac{\pi}{3}

We know that,

\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma

= 1

∴

\cos^2 \frac{\pi}{4} + \cos^2 \frac{\pi}{3} + \cos^2 \gamma

= 1

⇒

\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma

= 1

⇒

\frac{1}{2} + \frac{1}{4} + \cos^2 \gamma

= 1

⇒

\cos^2

γ =

\frac{1}{4}

⇒ cos γ =

\frac{1}{\sqrt{2}}

∴ γ =

\frac{\pi}{4}

Hence, direction cosines are cos α, cos β, cos γ i.e.

\frac{1}{\sqrt{2}},\ \frac{1}{2},\ \frac{1}{\sqrt{2}}

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