Question

Given that \( \vec{a} \parallel \vec{b} \), \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), and \( |\vec{a}| = 7 \), find \( |\vec{b}| \).

• A. \( |\vec{b}| = 7 \)
• B. \( |\vec{b}| = 14 \)
• C. \( |\vec{b}| = \frac{49}{7} \)
• D. \( |\vec{b}| = \frac{49}{14} \)

Hint:

When two vectors are parallel, their dot product is simply the product of their magnitudes. Use this property to solve for unknown magnitudes in problems involving parallel vectors.

Answer 1

The Correct Option is B

Given:
- \( \vec{a} \parallel \vec{b} \), meaning that vectors \( \vec{a} \) and \( \vec{b} \) are in the same direction. - \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), which is the dot product of vectors \( \vec{a} \) and \( \vec{b} \). - \( |\vec{a}| = 7 \), which is the magnitude of vector \( \vec{a} \).

1. Step 1: Use the formula for the dot product of two parallel vectors: The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) can be written as: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \] Since \( \vec{a} \parallel \vec{b} \), the angle \( \theta = 0^\circ \), and \( \cos(0^\circ) = 1 \). Thus, the dot product becomes: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \]

2. Step 2: Substitute the known values into the equation: We are given \( \vec{a} \cdot \vec{b} = \frac{49}{2} \) and \( |\vec{a}| = 7 \), so we substitute these into the equation: \[ \frac{49}{2} = 7 \cdot |\vec{b}| \]

3. Step 3: Solve for \( |\vec{b}| \): To find \( |\vec{b}| \), divide both sides by 7: \[ |\vec{b}| = \frac{49}{2 \cdot 7} = 14 \] Thus, the magnitude of \( \vec{b} \) is \( 14 \).