Question

A st. line which makes an angle of ${60^ \circ }$ with each of y and z-axes, is inclined with the z-axis at an angle
A) ${45^ \circ }$
B) ${30^ \circ }$
C) ${75^ \circ }$
D) ${60^ \circ }$

Answer 1

Hint:When the angle formed by the line with axes is given then that requires computing the cosines of the angles and setting the sum of the squares equal to one like ${\cos ^2}x + {\cos ^2}y + {\cos ^2}z = 1$.

Formula Used:
${\cos ^2}x + {\cos ^2}y + {\cos ^2}z = 1$

Complete step by step Solution:
The value of angle formed on y and z-axes is ${60^ \circ }$
The relationship between the angles formed by a vector with the x, y, and z axes are as follows in three dimensions.
Let the angle drawn on the x-axis be x, on the y-axis be y, and on the z-axis be z
Now, the direction cosine for the above angle will be $\cos x,\cos y,\cos z$ respectively.
${\cos ^2}x + {\cos ^2}y + {\cos ^2}z = 1$
According to the given information, we can say that
${\left( {\dfrac{1}{2}} \right)^2} + {\left( {\dfrac{1}{2}} \right)^2} + {\cos ^2}z = 1$
$\dfrac{2}{4} + {\cos ^2}z = 1$
$\dfrac{1}{2} + {\cos ^2}z = 1$
${\cos ^2}z = 1 - \dfrac{1}{2}$
${\cos ^2}z = \dfrac{1}{2}$
$ \Rightarrow \cos z = \dfrac{1}{{\sqrt 2 }}$
Thus the value of z is ${45^ \circ }$

Therefore, the correct option is A.

Note:As in this question there are angles formed by lines given we use the three-dimensional geometry to solve this with the help of direction cosine and trigonometry. Just knowing the basic formula will easily help to solve this type of question.