Question

Find the angle between unit vectors \(\mathbf{e_1}\) and \(\mathbf{e_2}\) if vectors \[ \mathbf{a} = \mathbf{e_1} + 2\mathbf{e_2},\quad \mathbf{b} = 5\mathbf{e_1} - 4\mathbf{e_2} \] are mutually perpendicular.

• A. \(\frac{\pi}{4}\)
• B. \(\frac{2\pi}{3}\)
• C. \(\frac{\pi}{2}\)
• D. \(\pi\)

Answer 1

The Correct Option is B

Use dot product: \[ \mathbf{a} \cdot \mathbf{b} = (e_1 + 2e_2) \cdot (5e_1 - 4e_2) = 5(e_1 \cdot e_1) - 4(e_1 \cdot e_2) + 10(e_2 \cdot e_1) - 8(e_2 \cdot e_2) \] \[ = 5 + 0 - 8 + 6(e_1 \cdot e_2) = 0 \Rightarrow 5 - 8 + 6 \cos \theta = 0 \Rightarrow -3 + 6 \cos \theta = 0 \Rightarrow \cos \theta = \frac{1}{2} \Rightarrow \theta = \frac{2\pi}{3} \]