Question

If $ |\vec{a}| = 5 $, $ |\vec{b}| = 8 $, $ |\vec{a} - \vec{b}| = 7 $, find the angle between } $ \vec{a} $ and $ \vec{b} $.

• A. \( 60^\circ \)
• B. \( 45^\circ \)
• C. \( 30^\circ \)
• D. \( 90^\circ \)

Hint:

To find the angle between two vectors, use the magnitude formula for the difference between vectors, and solve for the cosine of the angle.

Answer 1

The Correct Option is C

We are given: \[ |\vec{a}| = 5, \, |\vec{b}| = 8, \, |\vec{a} - \vec{b}| = 7 \] The formula for the magnitude of the difference of two vectors is: \[ |\vec{a} - \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 - 2 |\vec{a}| |\vec{b}| \cos \theta} \] Substituting the given values: \[ 7 = \sqrt{5^2 + 8^2 - 2 \times 5 \times 8 \times \cos \theta} \] Simplifying: \[ 7 = \sqrt{25 + 64 - 80 \cos \theta} \] \[ 49 = 89 - 80 \cos \theta \] \[ 80 \cos \theta = 40 \] \[ \cos \theta = \frac{1}{2} \]
Thus, the angle \( \theta \) is: \[ \theta = \cos^{-1} \left( \frac{1}{2} \right) = 60^\circ \]
Thus, the angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^\circ \).