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. \( \pi/2 \)
• C. \( \pi/3 \)
• D. None of the above

Hint:

If a vector \( \vec{u} \) is a scalar multiple of another vector \( \vec{v} \), i.e., \( \vec{u} = k \vec{v} \): - If \( k > 0 \), \( \vec{u} \) and \( \vec{v} \) are in the same direction (angle is 0). - If \( k < 0 \), \( \vec{u} \) and \( \vec{v} \) are in opposite directions (angle is \( \pi \)).

Answer 1

The Correct Option is A

Step 1: Understand the given relationships between the vectors. We are given three nonzero vectors a, b, and c such that: 1. \( a = 8b \) 2. \( c = -7b \) Step 2: Analyze the relationship between vectors a and b. Since \( a = 8b \) and 8 is a positive scalar, vector a is in the same direction as vector b. They are parallel (or collinear). The angle between a and b is 0. Step 3: Analyze the relationship between vectors c and b. Since \( c = -7b \) and -7 is a negative scalar, vector c is in the opposite direction to vector b. They are anti-parallel. The angle between c and b is \( \pi \) (180 degrees). Step 4: Determine the relationship between vectors a and c. From \( a = 8b \), we can express b in terms of a: \( b = \frac{1}{8}a \). Substitute this expression for b into the equation for c: \[ c = -7b = -7 \left( \frac{1}{8}a \right) \] \[ c = -\frac{7}{8}a \] Step 5: Determine the angle between a and c. Since \( c = -\frac{7}{8}a \) and \( -\frac{7}{8} \) is a negative scalar, vector c is in the opposite direction to vector a. They are anti-parallel. The angle between two anti-parallel vectors is \( \pi \) radians (180 degrees). Step 6: Compare the result with the given options. The angle between a and c is \( \pi \), which matches option (A).