Application on direction ratio and direction cosines of a vector

Direction Cosines of a Unit Vector

Question:

If a unit vector \( \hat{a} \) makes an angle \( \frac{\pi}{3} \) with \( \hat{i} \), \( \frac{\pi}{4} \) with \( \hat{j} \), and an acute angle \( \theta \) with \( \hat{k} \), find \( \theta \) and the direction cosines of \( \hat{a} \).

Solve:

A unit vector's direction cosines with the coordinate axes \( \hat{i}, \hat{j}, \hat{k} \) are given by:
\[ \cos \alpha, \cos \beta, \cos \gamma \] where \( \alpha, \beta, \gamma \) are the angles made with \( \hat{i}, \hat{j}, \hat{k} \) respectively.

Given:

• \( \alpha = \frac{\pi}{3} \Rightarrow \cos \alpha = \frac{1}{2} \)
• \( \beta = \frac{\pi}{4} \Rightarrow \cos \beta = \frac{1}{\sqrt{2}} \)
• \( \gamma = \theta \Rightarrow \cos \gamma = \cos \theta \)

For a unit vector:

\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

Substitute the values:

\[ \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1 \] \[ \frac{3}{4} + \cos^2 \theta = 1 \Rightarrow \cos^2 \theta = \frac{1}{4} \] \[ \cos \theta = \frac{1}{2} \quad (since \theta \; is \; acute) \] \[ \theta = \cos^{-1}\left( \frac{1}{2} \right) = \frac{\pi}{3} \]

Answer:

\( \theta = \frac{\pi}{3} \)
Direction cosines of \( \hat{a} \) are: \[ \left( \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2} \right) \]