Question

Let $\|.\|$ and $\langle .,.\rangle$ denote the standard norm and inner product in $\mathbb{R}^n$, respectively. If $u, v \in \mathbb{R}^3$ such that $\|u\| = \|v\| = 2$ and the angle between $u$ and $v$ is $\pi/3$, then

• A. $\|u - v\| = 2\sqrt{2}$
• B. $\langle u, v \rangle = 2\sqrt{3}$
• C. $\|u - v\| = 2\sqrt{3}$
• D. $\|u + v\| = 2\sqrt{3}$

Hint:

Use the identity $\|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2\|u\|\|v\|\cos\theta$ for vector magnitude problems involving angles.

Answer 1

The Correct Option is D

Step 1: Use norm and inner product relationship.
\[ \|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2\langle u, v \rangle \] Given $\|u\| = \|v\| = 2$ and $\langle u, v \rangle = \|u\|\|v\|\cos(\pi/3) = 4 \times \frac{1}{2} = 2$.
Step 2: Substitute values.
\[ \|u - v\|^2 = 4 + 4 - 2(2) = 4 \] \[ \|u - v\| = 2\sqrt{2} \]
Step 3: Conclusion.
Therefore, the correct answer is (A) $\|u - v\| = 2\sqrt{2}$.