Question

If a line makes the angles $\alpha ,\beta$ and $\gamma$ with the axes, then what is the value of $1+\cos 2\alpha +\cos 2\beta +\cos 2\gamma$ equal to ?

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is B $0$

Since, the line makes the angles $\alpha ,\beta$ and $\gamma$ with the axes

So, $\cos \alpha ,\cos \beta ,\cos \gamma$ will be the direction cosines of the line.

And we know that the sum of squares of the direction cosines of any line is $1$.

Now,

$1+\cos 2\alpha +\cos 2\beta +\cos 2\gamma$

$=1+(2{\cos }^2\alpha -1)+(2{\cos }^2\beta -1)+(2{\cos }^2\gamma -1)$

$=1+2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )-3$

$=1+2-3=0$

Hence, B is correct.