Question

The direction ratios of a straight line are \( 1, 3, 5 \). Then its direction cosines are:

• A. \( \frac{1}{\sqrt{9}}, \frac{3}{\sqrt{9}}, \frac{5}{\sqrt{9}} \)
• B. \( \frac{3}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}} \)
• C. \( \frac{5}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{1}{\sqrt{35}} \)
• D. none of these

Hint:

The direction cosines of a line can be found by dividing each of the direction ratios by the magnitude of the direction ratios.

Answer 1

The Correct Option is B

The direction cosines of a line are given by the formula: \[ \left( \frac{l}{\sqrt{l^2 + m^2 + n^2}}, \frac{m}{\sqrt{l^2 + m^2 + n^2}}, \frac{n}{\sqrt{l^2 + m^2 + n^2}} \right) \] where \( l, m, n \) are the direction ratios of the line. Given the direction ratios \( 1, 3, 5 \), we first calculate the magnitude of the direction ratios: \[ \text{Magnitude} = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35} \] Now, we calculate the direction cosines: \[ \text{Direction cosines} = \left( \frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}} \right) \] Thus, the correct answer is: \[ \boxed{\left( \frac{1}{\sqrt{35}}, \frac{3}{\sqrt{35}}, \frac{5}{\sqrt{35}} \right)} \]