Question

If \( a \), \( b \), \( c \) are nonzero vectors such that \( a = 8b \) and \( c = -7b \), then the angle between \( a \) and \( c \) is:

• A. \( \pi \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{3} \)
• D. None of the above

Hint:

The angle between two vectors is \( \pi \) when their dot product is negative and the vectors are in opposite directions.

Answer 1

The Correct Option is A

We are given the vectors: - \( a = 8b \) - \( c = -7b \) The angle between two vectors \( a \) and \( c \) can be found using the formula for the dot product: \[ \cos \theta = \frac{a \cdot c}{|a| |c|} \] First, calculate the dot product \( a \cdot c \). Since \( a = 8b \) and \( c = -7b \), we have: \[ a \cdot c = (8b) \cdot (-7b) = -56(b \cdot b) = -56|b|^2 \] Next, calculate the magnitudes of \( a \) and \( c \): \[ |a| = |8b| = 8|b| \quad \text{and} \quad |c| = |-7b| = 7|b| \] Now, substitute these values into the formula for \( \cos \theta \): \[ \cos \theta = \frac{-56|b|^2}{(8|b|)(7|b|)} = \frac{-56|b|^2}{56|b|^2} = -1 \] Thus, \( \cos \theta = -1 \), which means: \[ \theta = \pi \] Therefore, the angle between \( a \) and \( c \) is \( \boxed{\pi} \).