Question

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

• 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 determined by the formula: \[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}. \] The dot product \( \mathbf{a} \cdot \mathbf{b} \) is computed as: \[ \mathbf{a} \cdot \mathbf{b} = (2)(1) + (-1)(4) + (3)(-2) = 2 - 4 - 6 = -8. \] The magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \) are calculated as follows: \[ |\mathbf{a}| = \sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{4 + 1 + 9} = \sqrt{14}, \] \[ |\mathbf{b}| = \sqrt{1^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21}. \] Subsequently, \( \cos \theta \) is found to be: \[ \cos \theta = \frac{-8}{\sqrt{14} \times \sqrt{21}} = \frac{-8}{\sqrt{294}} \approx -0.462. \] Therefore, \( \theta \approx 60^\circ \). The final answer is: \[ \boxed{60^\circ}. \]