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

Step 1: Understanding the Question:
We need to find the direction cosines (\(l, m, n\)) of a line where the angles \(\alpha, \beta, \gamma\) with the X, Y, and Z axes are all equal.
Step 2: Key Formula or Approach:
Direction cosines satisfy the identity:
\[ l^2 + m^2 + n^2 = 1 \]
where \(l = \cos\alpha, m = \cos\beta, n = \cos\gamma\).
Step 3: Detailed Explanation:
Given that the line makes equal angles:
\(\alpha = \beta = \gamma\)
Taking cosine on both sides:
\(\cos\alpha = \cos\beta = \cos\gamma \Rightarrow l = m = n\)
Substitute this into the identity:
\[ l^2 + l^2 + l^2 = 1 \]
\[ 3l^2 = 1 \]
\[ l^2 = \frac{1}{3} \]
\[ l = \pm \frac{1}{\sqrt{3}} \]
Since \(l=m=n\), all direction cosines are \(\pm \frac{1}{\sqrt{3}}\).
Step 4: Final Answer:
The direction cosines are \( \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}} \).