Question

Let \( \vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is parallel to \( \vec{a} \) and \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), then \( |\vec{b}| = \)

• A. \(49 \)
• B. \(7 \)
• C. \(14 \)
• D. \( 7\sqrt{2} \)
• E. \( \frac{7}{2} \)

Hint:

For parallel vectors, angle = 0 so dot product simplifies to product of magnitudes.

Answer 1

Answer: (E)

Concept: If vectors are parallel: \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \]

Step 1: Find \( |\vec{a}| \).
\[ |\vec{a}| = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

Step 2: Use dot product.
\[ 7 \cdot |\vec{b}| = \frac{49}{2} \Rightarrow |\vec{b}| = \frac{7}{2} \]