Question

The product of the direction cosines of the line which makes equal angles with \[ox\], \[oy\], \[oz\] is
A. 1
B. \[\sqrt 3 \]
C. \[\dfrac{1}{{3\sqrt 3 }}\]
D. \[\sqrt 3 \]

Answer 1

Hint: First we will use the formula of direction cosines \[{\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1\], where \[\alpha \] is the angle with \[x\]–axis, \[\beta \] is the angle with \[y\]–axis and \[\gamma \] is the angle with \[z\]–axis. Then we will use that the line makes equal angles with three axis, here \[\alpha = \beta = \gamma \] and substitute the obtained value to find the direction cosines. Then we will find the product of the direction cosines of the line.

Complete step by step answer:

We are given that the direction cosines of the line which makes equal angles with \[ox\], \[oy\], \[oz\].

We know the formula of direction cosines will be used \[{\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1\] , where \[\alpha \] is the angle with \[x\]–axis, \[\beta \] is the angle with \[y\]–axis and \[\gamma \] is the angle with \[z\]–axis.

Since we are given that the line makes equal angles with three axes, here \[\alpha = \beta = \gamma \].

Thus, we have
\[
   \Rightarrow {\cos ^2}\alpha + {\cos ^2}\alpha + {\cos ^2}\alpha = 1 \\
   \Rightarrow 3{\cos ^2}\alpha = 1 \\
 \]

Dividing the above equation by 3 on both sides, we get
\[
   \Rightarrow \dfrac{{3{{\cos }^2}\alpha }}{3} = \dfrac{1}{3} \\
   \Rightarrow {\cos ^2}\alpha = \dfrac{1}{3} \\
 \]

Taking the square root of the above equation on both sides, we get
\[ \Rightarrow \cos \alpha = \pm \dfrac{1}{{\sqrt 3 }}\]

Since we know that the above value of direction cosine is the value of all the three direction cosines, so finding the product of the direction cosines of the line from the above equation, we get
\[
   \Rightarrow \left( {\cos \alpha } \right)\left( {\cos \alpha } \right)\left( {\cos \alpha } \right) \\
   \Rightarrow \pm \left( {\dfrac{1}{{\sqrt 3 }} \times \dfrac{1}{{\sqrt 3 }} \times \dfrac{1}{{\sqrt 3 }}} \right) \\
   \Rightarrow \pm \dfrac{1}{{3\sqrt 3 }} \\
 \]

Hence, option C is correct.

Note: In solving these types of questions, students must know that the direction cosines of a line, which are the cosines of the angles made by the line with positive directions of the co-ordinate axes. We can also convert the equation to \[\cos \beta \] or \[\cos \gamma \], the answer will be same. Students should also avoid calculation mistakes.