Question

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

Hint:

Remember the classic standard values for direction cosines: $\cos^2(45^\circ) + \cos^2(60^\circ) + \cos^2(60^\circ) = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1$. Recognizing these standard pythagorean relations can save you valuable calculation time during exams.

Answer 1

Concept: Direction cosines ($l, m, n$) of a vector are the cosines of the angles $\alpha, \beta, \gamma$ that the vector forms with the positive $x, y, z$ coordinate axes respectively.
• Direction Cosines Defined: $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$.
• Fundamental Identity: For any vector, the sum of the squares of its direction cosines is always equal to unity: $l^2 + m^2 + n^2 = 1$.
• Unit Vector Representation: Any unit vector $\vec{a}$ can be expressed in terms of its direction cosines as $\vec{a} = l\hat{i} + m\hat{j} + n\hat{k}$.

Step 1: Substitute the given angles into the direction cosine identity to determine the angle $\theta$.
The given angles are: \[ \alpha = \frac{\pi}{4}, \quad \beta = \frac{\pi}{3}, \quad \gamma = \theta \] Computing individual direction cosines: \[ l = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \] \[ m = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \] \[ n = \cos\theta \] Substitute these into the identity $l^2 + m^2 + n^2 = 1$: \[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2\theta = 1 \] \[ \frac{1}{2} + \frac{1}{4} + \cos^2\theta = 1 \] Combine the fractions: \[ \frac{2 + 1}{4} + \cos^2\theta = 1 \quad \implies \quad \frac{3}{4} + \cos^2\theta = 1 \] Isolate $\cos^2\theta$: \[ \cos^2\theta = 1 - \frac{3}{4} = \frac{1}{4} \] Taking the square root on both sides: \[ \cos\theta = \pm \sqrt{\frac{1}{4}} = \pm \frac{1}{2} \] Since the problem explicitly specifies that $\theta$ is an acute angle, $\cos\theta$ must be strictly positive. Therefore: \[ \cos\theta = \frac{1}{2} \quad \implies \quad \theta = \frac{\pi}{3} \]

Step 2: Formulate the components of the unit vector $\vec{a}$.
The direction cosines are $l = \frac{1}{\sqrt{2}}$, $m = \frac{1}{2}$, and $n = \frac{1}{2}$. Since $\vec{a}$ is a unit vector, its components along the axes are exactly its direction cosines: \[ \vec{a} = l\hat{i} + m\hat{j} + n\hat{k} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{2}\hat{j} + \frac{1}{2}\hat{k} \] Thus, the vector components are $\left(\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}\right)$.