Question

If \( |\mathbf{a} - \mathbf{b}| = \frac{\sqrt{3}}{2} \) where \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).

Hint:

To find the angle between two vectors, first compute their dot product and then take the inverse cosine of the result.

Answer 1

Step 1: Use the given equation.
We are given that \( |\mathbf{a} - \mathbf{b}| = \frac{\sqrt{3}}{2} \). Using the formula for the magnitude of the difference of two vectors: \[ |\mathbf{a} - \mathbf{b}| = \sqrt{|\mathbf{a}|^2 + |\mathbf{b}|^2 - 2 \mathbf{a} \cdot \mathbf{b}} \] Since \( \mathbf{a} \) and \( \mathbf{b} \) are unit vectors, \( |\mathbf{a}| = |\mathbf{b}| = 1 \), so the equation simplifies to: \[ |\mathbf{a} - \mathbf{b}| = \sqrt{2 - 2 \mathbf{a} \cdot \mathbf{b}} \]
Step 2: Substitute the given magnitude.
Substitute \( |\mathbf{a} - \mathbf{b}| = \frac{\sqrt{3}}{2} \) into the equation: \[ \frac{\sqrt{3}}{2} = \sqrt{2 - 2 \mathbf{a} \cdot \mathbf{b}} \] Square both sides: \[ \frac{3}{4} = 2 - 2 \mathbf{a} \cdot \mathbf{b} \]
Step 3: Solve for \( \mathbf{a} \cdot \mathbf{b} \).
Rearrange the equation: \[ 2 \mathbf{a} \cdot \mathbf{b} = 2 - \frac{3}{4} = \frac{5}{4} \] So: \[ \mathbf{a} \cdot \mathbf{b} = \frac{5}{8} \]
Step 4: Find the angle between \( \mathbf{a} \) and \( \mathbf{b} \).
The dot product of two vectors is given by: \[ \mathbf{a} \cdot \mathbf{b} = \cos \theta \] Thus: \[ \cos \theta = \frac{5}{8} \] Therefore, the angle \( \theta \) is: \[ \theta = \cos^{-1} \left( \frac{5}{8} \right) \] Thus, the angle between \( \mathbf{a} \) and \( \mathbf{b} \) is: \[ \boxed{\cos^{-1} \left( \frac{5}{8} \right)} \]