Question

The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:

• A. 48 and 16
• B. 3 and 1
• C. 24 and 8
• D. 6 and 2

Hint:

When working with vector magnitudes and dot products, remember to use vector identities like \( (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = \vec{a}^2 - \vec{b}^2 \).

Answer 1

The Correct Option is C

First, simplify the given equation using the identity for the dot product: \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = \vec{a}^2 - \vec{b}^2. \] Thus, the equation becomes: \[ \vec{a}^2 - \vec{b}^2 = 512. \] Next, we use the condition \( |\vec{a}| = 3 |\vec{b}| \). Let \( |\vec{b}| = x \), so \( |\vec{a}| = 3x \). Therefore, we can rewrite the equation as: \[ (3x)^2 - x^2 = 512 \quad \Rightarrow \quad 9x^2 - x^2 = 512 \quad \Rightarrow \quad 8x^2 = 512. \] Solving for \( x \): \[ x^2 = \frac{512}{8} = 64 \quad \Rightarrow \quad x = 8. \] Hence, \( |\vec{b}| = 8 \) and \( |\vec{a}| = 3 \times 8 = 24 \).