Question

Find the direction cosines of a line that makes equal angles with the coordinate axes.

• A. \( \pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}, 0 \)
• B. \( \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}} \)
• C. \( \pm \frac{1}{2}, \pm \frac{1}{2}, \pm \frac{1}{2} \)
• D. \( \pm 1, 0, 0 \)

Hint:

Direction cosines always satisfy \[ l^2+m^2+n^2=1 \] If a line makes equal angles with the coordinate axes, then \(l=m=n\).

Answer 1

The Correct Option is B

Concept: If a line makes angles \( \alpha, \beta, \gamma \) with the \(x\)-, \(y\)- and \(z\)-axes respectively, then its direction cosines are \[ l=\cos\alpha,\quad m=\cos\beta,\quad n=\cos\gamma \] They satisfy the identity: \[ l^2+m^2+n^2=1 \]

Step 1: Use the condition of equal angles. If the line makes equal angles with the coordinate axes, then \[ \alpha=\beta=\gamma \] Thus, \[ l=m=n \]

Step 2: Substitute into the identity. \[ l^2+m^2+n^2=1 \] \[ l^2+l^2+l^2=1 \] \[ 3l^2=1 \] \[ l=\pm \frac{1}{\sqrt{3}} \]

Step 3: Determine the direction cosines. Since \(l=m=n\), \[ l=m=n=\pm \frac{1}{\sqrt{3}} \] Thus, the direction cosines are \[ \boxed{\pm \frac{1}{\sqrt{3}},\ \pm \frac{1}{\sqrt{3}},\ \pm \frac{1}{\sqrt{3}}} \]