Question

The direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are

• A. $\left(x_{2} - x_{1}\right)$, $\left(y_{2} - y_{1}\right)$, $\left(z_{2} - z_{1}\right)$
• B. $\left(x_{2} + x_{1}\right)$, $\left(y_{2} + y_{1}\right)$, $\left(z_{2} + z_{1}\right)$
• C. $\frac{x_{2}-x_{1}}{PQ}$, $\frac{y_{2}-y_{1}}{PQ}$, $\frac{z_{2}-z_{1}}{PQ}$
• D. $\frac{x_{2}+x_{1}}{PQ}$, $\frac{y_{2}+y_{1}}{PQ}$, $\frac{z_{2}+z_{1}}{PQ}$

Answer 1

Answer: (C)

$P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$
$\therefore $ Direction ratios of line $PQ=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right)$
$\Rightarrow $ direction cosine of $PQ =$
$\bigg[\frac{x_{2}-x_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}$,
$\frac{y_{2}-y_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}$,
$\frac{z_{2}-z_{1}}{\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}\bigg]$
$=\left[\frac{x_{2}-x_{1}}{PQ}, \frac{y_{2}-y_{1}}{PQ}, \frac{z_{2}-z_{1}}{PQ}\right]$
where $PQ=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$