Question

If \( |\vec{a}| = 2 \) and \( -3 \leq k \leq 2 \), then \( |k\vec{a} | \in: \)

• A. \( [-6, 4] \)
• B. \( [0, 4] \)
• C. \( [4, 6] \)
• D. \( [0, 6] \)

Hint:

The maximum value of a dot product is achieved when the vectors are parallel, and the minimum is achieved when they are perpendicular.

Answer 1

The Correct Option is D

Step 1: {Recall the dot product formula}
The dot product of two vectors \( \vec{a} \) and \( \vec{k} \) is: \[ |\vec{a} \cdot \vec{k}| = |\vec{a}| |\vec{k}| \cos \theta, \] where \( \theta \) is the angle between the vectors. 
Step 2: {Determine the range of \( |\vec{a} \cdot \vec{k}| \)}
Since \( |\vec{a}| = 2 \), the magnitude of \( \vec{k} \) varies as: \[ |\vec{k}| \in [-3, 2]. \] The maximum value of \( |\vec{a} \cdot \vec{k}| \) occurs when \( \cos \theta = 1 \): \[ |\vec{a} \cdot \vec{k}|_{{max}} = 2 \cdot 3 = 6. \] The minimum value of \( |\vec{a} \cdot \vec{k}| \) occurs when \( \cos \theta = 0 \): \[ |\vec{a} \cdot \vec{k}|_{{min}} = 0. \] 
Step 3: {Conclude the result}
Thus, \( |\vec{a} \cdot \vec{k}| \in [0, 6] \).