# Write the direction cosines of the vector $$-2\hat i + \hat j - 5\hat k$$.

Toolbox:
• The cosines of the angle made by the vector with the coordinate axes is called direction cosines
• Direction cosine (D.C) of vector $x\hat i + y\hat j + 2\hat k$ is $\large\frac{x}{\sqrt{x^2+y^2+z^2}}, \large\frac{y}{\sqrt{x^2+y^2+z^2}}, \large\frac{z}{\sqrt{x^2+y^2+z^2}}$
Step 1:
Let $\overrightarrow a=-2\hat i+\hat j-5\hat k$
$\mid \overrightarrow a\mid=\sqrt{(-2)^2+(1)^2+(-5)^2}$
$\qquad=\sqrt{4+1+25}$
$\qquad=\sqrt{30}$
Step 2:
Direction cosine (D.C) of vector $x\hat i + y\hat j + 2\hat k$ is $\large\frac{x}{\sqrt{x^2+y^2+z^2}}, \large\frac{y}{\sqrt{x^2+y^2+z^2}}, \large\frac{z}{\sqrt{x^2+y^2+z^2}}$
Hence the direction cosines are $\large\frac{-2}{\sqrt{30}},\frac{1}{\sqrt{30}},\frac{-5}{\sqrt{30}}$