Question

A vector $\vec p$ is making $45^\circ$ and $60^\circ$ with positive $x$ and $y$ axes respectively. Then the angle made this vector with positive $z$ axis is:
A. $255^\circ$
B. $75^\circ$
C. $30^\circ$
D. $60^\circ$

Answer 1

First of all, we will pick out the angles made with the two axes given. Then we will use the formula of direction cosines to find the other angle. We will substitute the required values and manipulate accordingly to obtain the result.

Complete step by step answer:
In the given problem, we are supplied the following data:
The vector is given as $\vec p$ which makes different angles with the respective axes.
The angles made by the respective vector with the axes are $45^\circ$ and $60^\circ$ .
We are asked to find the angle made by this vector with the positive $z$ axis.
To solve this problem, we will use a formula, which gives the direction cosines, with the help of the angles represented by $\alpha$, $\beta$ and $\gamma$ is as follows:
${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$ …… (1)
Where,
$\alpha$ indicates the angle made with the positive $x$ axis.
$\beta$ indicates the angle made with the positive $y$ axis.
$\gamma$ indicates the angle made with the positive $z$ axis.
Now, we substitute the required values in the equation (1) and we get:
{\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1 \\\ {\cos ^2}45^\circ + {\cos ^2}60^\circ + {\cos ^2}\gamma = 1 \\\ {\left( {\dfrac{1}{{\sqrt 2 }}} \right)^2} + {\left( {\dfrac{1}{2}} \right)^2} + {\cos ^2}\gamma = 1 \\\ \dfrac{1}{2} + \dfrac{1}{4} + {\cos ^2}\gamma = 1 \\\
Again, we manipulate further:
{\cos ^2}\gamma = 1 - \dfrac{1}{2} - \dfrac{1}{4} \\\
\implies {\cos ^2}\gamma = \dfrac{1}{4} \\\
\implies \cos \gamma = \dfrac{1}{2} \\\
\implies \gamma = {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right) \\\
In the last step, we will find the angle which is given by the inverse cosine of that fraction:
$\gamma = 60^\circ$
Hence, the angle made by this vector with a positive $z$ axis is $60^\circ$ .

So, the correct answer is “Option D”.

Note:
This problem is based on the concept of direction cosines. This is an arrangement in three-dimensional space. The direction cosines represent the cosines of the angles made between the vector and the three coordinate directions.