Question

A unit vector \( \vec{a} \) is such that it makes an angle \( \frac{\pi}{4} \) with x-axis, \( \frac{\pi}{3} \) with y-axis and an acute angle \( \theta \) with z-axis. Find \( \theta \) and the components of \( \vec{a} \).

Hint:

"Acute angle" means \( 0 \le \theta < \pi/2 \), so the cosine is positive. "Obtuse" would mean the cosine is negative.
For a unit vector, the sum of the squares of its components must always be 1.

Answer 1

Concept:
• Let \( \alpha, \beta, \gamma \) be the angles made by a vector with the x, y, and z axes respectively.
• The direction cosines are \( l = \cos \alpha, m = \cos \beta, n = \cos \gamma \).
• Fundamental property: \( l^2 + m^2 + n^2 = 1 \).
• For a unit vector \( \vec{a} \), its components are simply \( (l, m, n) \).

Step 1: Calculate the direction cosines \( l \) and \( m \)
Given \( \alpha = \frac{\pi}{4} \) and \( \beta = \frac{\pi}{3} \).
\( l = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \)
\( m = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

Step 2: Solve for \( n \) and the angle \( \theta \)
Using \( l^2 + m^2 + n^2 = 1 \):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + n^2 = 1 \] \[ \frac{1}{2} + \frac{1}{4} + n^2 = 1 \] \[ \frac{3}{4} + n^2 = 1 \implies n^2 = 1 - \frac{3}{4} = \frac{1}{4} \] Since \( \theta \) (which is \( \gamma \)) is acute, \( n = \cos \theta \) must be positive:
\[ n = \frac{1}{2} \implies \cos \theta = \frac{1}{2} \implies \theta = \frac{\pi}{3} \]

Step 3: State the components of the unit vector \( \vec{a} \)
The components of the unit vector are \( (l, m, n) \):
\[ \vec{a} = l\hat{i} + m\hat{j} + n\hat{k} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{2}\hat{j} + \frac{1}{2}\hat{k} \]