Question

Find the angle between the vectors \( \mathbf{a} = (2, 3, 1) \) and \( \mathbf{b} = (1, -1, 4) \).

• A. \( 45^\circ \)
• B. \( 60^\circ \)
• C. \( 90^\circ \)
• D. \( 120^\circ \)

Hint:

Use the dot product and magnitudes of vectors to find the angle between them.

Answer 1

The Correct Option is B

The angle \( \theta \) between two vectors is given by: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}. \] First, calculate the dot product \( \mathbf{a} \cdot \mathbf{b} \): \[ \mathbf{a} \cdot \mathbf{b} = 2 \times 1 + 3 \times (-1) + 1 \times 4 = 2 - 3 + 4 = 3. \] Next, calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \): \[ |\mathbf{a}| = \sqrt{2^2 + 3^2 + 1^2} = \sqrt{4 + 9 + 1} = \sqrt{14}, \] \[ |\mathbf{b}| = \sqrt{1^2 + (-1)^2 + 4^2} = \sqrt{1 + 1 + 16} = \sqrt{18}. \] Now, calculate \( \cos \theta \): \[ \cos \theta = \frac{3}{\sqrt{14} \times \sqrt{18}} = \frac{3}{\sqrt{252}} \approx 0.188. \] Thus, \( \theta = \cos^{-1}(0.188) \approx 60^\circ \). The correct answer is: \[ \boxed{60^\circ}. \]