Question

If $|\vec{u}| = 2$ and $\vec{u}$ makes angles of $60^\circ$ and $120^\circ$ with the axes $OX$ and $OY$ at the origin, then $\vec{u} =$

• A. $\hat{i} + \hat{j} + 2\hat{k}$
• B. $2(\hat{i} + \hat{j} \pm \sqrt{2}\hat{k})$
• C. $2(\hat{i} - \hat{j} + \sqrt{2}\hat{k})$
• D. $\hat{i} - \hat{j} \pm \sqrt{2}\hat{k}$

Hint:

You can use option elimination to find the answer without calculating the third angle! Compute the first two components directly: $u_x = |\vec{u}|\cos(60^\circ) = 2(0.5) = 1$, and $u_y = |\vec{u}|\cos(120^\circ) = 2(-0.5) = -1$. Looking at the choices, only option (D) features $+1\hat{i} - 1\hat{j}$ at the front, saving you valuable time during the test!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A 3D vector has magnitude |ū| = 2, makes α = 60° with the X-axis and β = 120° with the Y-axis. Find its component form.

Step 2: Key Formula or Approach:
Direction cosines are l = cos α, m = cos β, n = cos γ. They satisfy l² + m² + n² = 1. The vector is then ū = |ū|(l î + m ĵ + n k̂).

Step 3: Detailed Explanation:
Compute the known cosines: l = cos 60° = 1/2, m = cos 120° = -1/2. Substitute into the identity: (1/2)² + (-1/2)² + n² = 1 → 1/4 + 1/4 + n² = 1 → 1/2 + n² = 1 → n² = 1/2 → n = ±1/√2. With |ū| = 2, the vector is ū = 2(½î - ½ĵ ± (1/√2)k̂) = î - ĵ ± (2/√2)k̂ = î - ĵ ± √2 k̂.

Step 4: Final Answer:
The vector is î - ĵ ± √2 k̂, option (D).