Question

If \( \alpha, \beta, \gamma \) are the angles made by \[ \frac{x-1}{3} = \frac{y+1}{2} = -\frac{z}{1} \text{ with the coordinate axes, then } \] \((\cos\alpha, \cos\beta, \cos\gamma) = \)

• A. \( \left( \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}} \right) \)
• B. \( \left( \frac{3}{\sqrt{7}}, \frac{-2}{\sqrt{7}}, \frac{-1}{\sqrt{7}} \right) \)
• C. \( \left( \frac{3}{\sqrt{14}}, \frac{-2}{\sqrt{14}}, \frac{-1}{\sqrt{14}} \right) \)
• D. \( \left( \frac{3}{\sqrt{7}}, \frac{3}{\sqrt{7}}, \frac{-1}{\sqrt{7}} \right) \)
• E. \( \left( \frac{-3}{\sqrt{14}}, \frac{-2}{\sqrt{14}}, \frac{-1}{\sqrt{14}} \right) \)

Hint:

The direction cosines are found by dividing the direction ratios of the line by the magnitude of the direction ratios. Always compute the magnitude first, then divide each direction ratio by the magnitude to find the direction cosines.

Answer 1

The Correct Option is A

We are given the equation: \[ \frac{x-1}{3} = \frac{y+1}{2} = -\frac{z}{1}. \] This equation represents a direction cosines relationship. The general equation for direction cosines is: \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c}, \] where \(a\), \(b\), and \(c\) represent the direction ratios of the line, and \(x\), \(y\), and \(z\) are the coordinates. Comparing the given equation with this form, we get: \[ \frac{x-1}{3} = \frac{y+1}{2} = -\frac{z}{1}. \] Thus, the direction ratios are: \[ a = 3, \quad b = 2, \quad c = -1. \] The direction cosines are given by: \[ \cos\alpha = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad \cos\beta = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad \cos\gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}}. \] Step 1: Calculate the magnitude of the direction ratios: \[ \sqrt{a^2 + b^2 + c^2} = \sqrt{3^2 + 2^2 + (-1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14}. \] Step 2: Calculate the direction cosines: \[ \cos\alpha = \frac{3}{\sqrt{14}}, \quad \cos\beta = \frac{2}{\sqrt{14}}, \quad \cos\gamma = \frac{-1}{\sqrt{14}}. \] Thus, the correct answer is option (A): \[ (\cos\alpha, \cos\beta, \cos\gamma) = \left( \frac{3}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}} \right). \] Therefore, the correct answer is option (A).