Vector Algebra
Angle between Vectors
Grade 12
Question:
<p>If <strong>a</strong> + <strong>b</strong> + <strong>c</strong> = 0 and |<strong>a</strong>| = 3, |<strong>b</strong>| = 5, |<strong>c</strong>| = 7, then the angle between <strong>a</strong> and <strong>b</strong> is</p>
<p>(a) \(\frac{\pi}{2}\)</p>
<p>(b) \(\frac{\pi}{3}\)</p>
<p>(c) \(\frac{\pi}{4}\)</p>
<p>(d) \(\frac{\pi}{6}\)</p>
Step-by-Step Solution
Key Concept: Use the law of cosines with the constraint that a + b + c = 0 to find the angle between vectors.
Step 1: Let \(\theta\) be the angle between a and b . Then, \(\angle C = \pi - \theta\). Step 2: Apply the law of cosines: \(\cos(\pi - \theta) = \frac{3^2 + 5^2 - 7^2}{2(3)(5)}\) Step 3: \(-\cos\theta = \frac{9 + 25 - 49}{30} = \frac{-15}{30} = -\frac{1}{2}\) Step 4: \(\cos\theta = \frac{1}{2}\) \(\Rightarrow\) \(\theta = 60° = \frac{\pi}{3}\) ∴ Answer is (b).
Correct Answer: B