A line AB in three-dimensional space makes angles {45^ \circ } and {120^ \circ } with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle heta with the positive z-axis, then heta equals :

Question

A line AB in three-dimensional space makes angles {45^ \circ } and {120^ \circ } with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle 	heta with the positive z-axis, then 	heta equals :

Accepted Answer

Answer: B. {60^ \circ } Direction cosines of the line : \ell = \cos {45^ \circ } = {1 \over {\sqrt 2 }},m = \cos {120^ \circ } = {{ - 1} \over 2},\pi = \cos 	heta where 	heta is the angle, which line makes with positive z-axis. Now {\ell ^2} + {m^2} + {n^2} = 1 \Rightarrow {1 \over 2} + {1 \over 4} + {\cos ^2}	heta = 1,\,\,{\cos ^2}	heta = {1 \over 4} \Rightarrow \cos 	heta = {1 \over 2}\,\,\,\,\left( 	heta ight. being acute) \Rightarrow 0 = {\pi \over 3}

A line $$AB$$ in three-dimensional space makes angles $${45^ \circ }$$ and $${120^ \circ }$$ with the positive $$x$$-axis and the positive $$y$$-axis respectively. If $$AB$$ makes an acute angle $$\theta $$ with the positive $$z$$-axis, then $$\theta $$ equals :

Answer

Explanation

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