Question

A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. If the angle $\alpha$ which the line makes with the positive direction of x-axis is acute, the direction cosines of the line are

• A. $\frac{2}{3},\frac{-2}{3},\frac{-1}{3}$
• B. $\frac{2}{3},\frac{2}{3},\frac{-1}{3}$
• C. $\frac{2}{3},\frac{-2}{3},\frac{1}{3}$
• D. $\frac{2}{3},\frac{2}{3},\frac{1}{3}$

Answer 1

The correct option is A $\frac{2}{3},\frac{-2}{3},\frac{-1}{3}$

Let $\overrightarrow{OA}=6\hat{i}-7\hat{j}-\hat{k}$

$\overrightarrow{OB}=2\hat{i}-3\hat{j}+\hat{k}$

$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

= $(6\hat{i}-7\hat{j}-\hat{k})-(2\hat{i}-3\hat{j}+\hat{k})$

= $4\hat{i}-4\hat{j}-2\hat{k}$

|$\overrightarrow{AB}$| = $\sqrt{(4{)}^2+(-4{)}^2+(-2{)}^2}$

= $\sqrt{16+16+4}$

= $\sqrt{36}$ $\Rightarrow 6$

The direction cosines are : $(\frac{4}{6},\frac{-4}{6},\frac{-2}{6})$ $÷2$.

$\Rightarrow$ $\frac{2}{3},\frac{-2}{3},\frac{-1}{3}$.