Question

If direction ratios of a line are -18, 12, -4, then find its direction cosines.

Hint:

Direction cosines are calculated by dividing the direction ratios by the magnitude of the direction ratios.

Answer 1

Step 1: Understand the relation between direction ratios and direction cosines.
The direction ratios of a line are proportional to the direction cosines. Let the direction ratios be \( a = -18 \), \( b = 12 \), and \( c = -4 \). The direction cosines \( \alpha \), \( \beta \), and \( \gamma \) are given by the following relations: \[ \alpha = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \beta = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \gamma = \frac{c}{\sqrt{a^2 + b^2 + c^2}}. \]

Step 2: Calculate the magnitude of the direction ratios.
First, calculate the magnitude of the direction ratios: \[ \text{Magnitude} = \sqrt{(-18)^2 + 12^2 + (-4)^2} = \sqrt{324 + 144 + 16} = \sqrt{484} = 22. \]

Step 3: Calculate the direction cosines.
Now, calculate the direction cosines: \[ \alpha = \frac{-18}{22} = -\frac{9}{11}, \beta = \frac{12}{22} = \frac{6}{11}, \gamma = \frac{-4}{22} = -\frac{2}{11}. \]

Step 4: Conclusion.
Thus, the direction cosines of the line are \( \alpha = -\frac{9}{11}, \beta = \frac{6}{11}, \gamma = -\frac{2}{11} \).