# If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Question:

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Solution:

If a line has direction ratios of −18, 12, and −4, then its direction cosines are

$\frac{-18}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}, \frac{12}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}, \frac{-4}{\sqrt{(-18)^{2}+(12)^{2}+(-4)^{2}}}$

i.e., $\frac{-18}{22}, \frac{12}{22}, \frac{-4}{22}$

$\frac{-9}{11}, \frac{6}{11}, \frac{-2}{11}$

Thus, the direction cosines are $-\frac{9}{11}, \frac{6}{11}$, and $\frac{-2}{11}$.