Question

If \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \) and \( |\vec{a}| = 10|\vec{b}| \), then :

• A. \( |\vec{a}| = \sqrt{2} \)
• B. \( |\vec{b}| = \sqrt{2} \)
• C. \( |\vec{b}| = 10\sqrt{2} \)
• D. \( |\vec{a}| = \frac{10}{\sqrt{2}} \)

Hint:

The expression \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = |\vec{a}|^2 - |\vec{b}|^2 \) is the vector equivalent of the algebraic difference of squares. Always simplify dot products algebraically before substituting numerical values to avoid unnecessary computation.

Answer 1

The Correct Option is B

Concept:
• Distributive Property: The dot product distributes over addition and subtraction: \( (\vec{u} + \vec{v}) \cdot \vec{w} = \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w} \).
• Self-Dot Product: The dot product of a vector with itself is the square of its magnitude: \( \vec{a} \cdot \vec{a} = |\vec{a}|^2 \).
• Commutativity: Dot product is commutative: \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \).

Step 1: Expand the dot product expression
Given \( (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 198 \). Expanding using the distributive property: \[ \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} = 198 \] Since \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \), the middle terms cancel: \[ |\vec{a}|^2 - |\vec{b}|^2 = 198 \]

Step 2: Substitute the relationship between magnitudes
We are given \( |\vec{a}| = 10|\vec{b}| \). Squaring both sides: \[ |\vec{a}|^2 = (10|\vec{b}|)^2 = 100|\vec{b}|^2 \] Substitute this into the expanded equation: \[ 100|\vec{b}|^2 - |\vec{b}|^2 = 198 \] \[ 99|\vec{b}|^2 = 198 \]

Step 3: Solve for the magnitudes
Divide by 99: \[ |\vec{b}|^2 = \frac{198}{99} = 2 \] Taking the square root: \[ |\vec{b}| = \sqrt{2} \] Using \( |\vec{a}| = 10|\vec{b}| \), we also find \( |\vec{a}| = 10\sqrt{2} \).