Vector Algebra
Multiple Correct – Vector Conditions
Grade 12
Question:
<p>Let \(\hat{a}\) and \(\hat{b}\) be unit vectors and \(\theta\) be the angle
between them. Then \(\hat{a}+\hat{b}\) is a unit vector if and only if:</p>
\(\theta=\dfrac{\pi}{3}\)
\(\theta=\dfrac{2\pi}{3}\)
\(\theta=\dfrac{\pi}{2}\)
\(\theta=\pi\)
Step-by-Step Solution
Key Concept: For |a + b| = 1: expand |a + b|^2 = 1. This gives a \cdot b = -1/2, so cos \theta = -1/2.
\(|\hat{a}+\hat{b}|^2=|\hat{a}|^2+2\hat{a}\cdot\hat{b}+|\hat{b}|^2=1+2\cos\theta+1=2+2\cos\theta\).
Setting \(=1\): \(2\cos\theta=-1\Rightarrow\cos\theta=-\tfrac{1}{2}\Rightarrow\theta=\dfrac{2\pi}{3}\).
Answer: B .
Correct Answer: B