1.

A line makes an angle of 30° and 60° with the positive direction of X and Y-axis. Find the angle formed by the line with the positive direction of Z-axis.1. 60°2. 30°3. 45°4. 90°5. 140°

Answer» Correct Answer - Option 4 : 90°

Concept:

The direction cosines of a vector are the cosines of the angles between the vector and the three +ve coordinate axes i.e. X, Y, Z axes.

 l2 + m+ n2 = 1

where l = the cosines of the angles between the vector and  X-axis.

m = the cosines of the angles between the vector and  Y-axis.

n = the cosines of the angles between the vector and  Z-axis.

Calculation:

Let the line makes an angle γ with the positive direction of Z-axis. Thus it makes angle 30°, 60° and γ with the three axes.

∴ the d.c's of line are cos 30°, cos 60°, cos γ i. e. \(\dfrac{\sqrt{3}}{2}\)\(\dfrac{1}{2}\), cos γ

we know that, l2 + m2 + n2 = 1

∴ \({\left( {\sqrt 3 /2} \right)^2} + {\left( {1/2} \right)^2} + {\left( {\cos γ } \right)^2} = 1\)

or cos2 γ = 1 - 1

⇒ cos2 γ = 0

⇒ cos γ = 0

or, γ = 90°

Thus the line makes an angle of 90° with the Z-axis.



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