Vectors
Unit Vectors
MMTS_Full_Test_21
Grade 12
Question:
$\hat{a}$ and $\hat{b}$ are two unit vectors and $\theta$ is the angle between them. Then $|\hat{a}+\hat{b}|^2-|\hat{a}-\hat{b}|^2=$
$2\sin\theta$
$2\cos\theta$
$4\sin\theta$
$4\cos\theta$
Step-by-Step Solution
Key Concept: $(\hat{a}+\hat{b})\cdot(\hat{a}+\hat{b})-(\hat{a}-\hat{b})\cdot(\hat{a}-\hat{b})=4\hat{a}\cdot\hat{b}=4\cos\theta$
$|\hat{a}+\hat{b}|^2=2+2\cos\theta$; $|\hat{a}-\hat{b}|^2=2-2\cos\theta$. Difference $=4\cos\theta$.
Correct Answer: 4