Question

A line AB in three dimensional space making an angle ${45^\circ }$ and ${120^\circ }$ with the positive x-axis and the positive y-axis respectively . If AB makes an acute angle e with the positive z-axis , then e equals to :
A) ${45^\circ }$
B) ${60^\circ }$
C) ${75^\circ }$
D) ${30^\circ }$

Answer 1

Hint: As we know that if the line AB makes an angle (a , b ,c ) with the positive (x , y , z) axis then ${\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1$ hence in this question two angles are know third one is find out from this formula .

Complete step-by-step answer:
In the question it is given that Line AB is in three dimensional making angles ${45^\circ }$ and ${120^\circ }$ with the positive x-axis and the positive y-axis respectively .
And makes an acute angle e with the positive z-axis mean that angle e < ${90^\circ }$ .
Hence from the Vector property we know that if the line AB makes an angle (a , b ,c ) with the positive (x , y , z) axis then ${\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1$ .
Hence from the question a = ${45^\circ }$ , b = ${120^\circ }$and c = e ,
${\cos ^2}{45^\circ } + {\cos ^2}{120^\circ } + {\cos ^2}e = 1$
${\cos ^2}e = 1 - ({\cos ^2}{45^\circ } + {\cos ^2}{120^\circ })$
${\cos ^2}e = 1 - {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2} - {\left( {\dfrac{{ - 1}}{2}} \right)^2}$
${\cos ^2}e = 1 - \left( {\dfrac{1}{2}} \right) - \left( {\dfrac{1}{4}} \right)$
${\cos ^2}e = 1 - \left( {\dfrac{3}{4}} \right)$
${\cos ^2}e = \left( {\dfrac{1}{4}} \right)$
$\cos e = \pm \left( {\dfrac{1}{2}} \right)$
For the acute angle e < ${90^\circ }$ hence only positive will be considered .
$\cos e = \left( {\dfrac{1}{2}} \right)$
e = ${60^\circ }$

Hence option B will be the correct answer.

Note: In this question it is given in that the angle is e is acute angle , If it is obtuse angle then we have to consider the negative value of $\cos e$that is $\cos e = - \left( {\dfrac{1}{2}} \right)$ Hence the $e = {120^\circ }$ .
Multiplication of a vector by a scalar quantity is called Scaling. In this type of multiplication, only the magnitude of a vector is changed not the direction.
Always remember that if a line makes an angle (a , b ,c ) with the positive (x , y , z) axis then ${\cos ^2}a + {\cos ^2}b + {\cos ^2}c = 1$ .