Question

If $ | \vec{a} | = 3 $, $ | \vec{b} | = 2 $, then find $ (3\vec{a} - 2\vec{b}) \cdot (3\vec{a} + 2\vec{b}) $.

• A. \( 27 \)
• B. \( 0 \)
• C. \( 15 \)
• D. \( 25 \)

Hint:

Remember that when simplifying dot products involving vector operations, the distributive property is very useful. Be mindful of the squared magnitudes of the vectors when calculating the dot product of a vector with itself.

Answer 1

The Correct Option is D

We are given \( | \vec{a} | = 3 \) and \( | \vec{b} | = 2 \). We are asked to find: \[ (3\vec{a} - 2\vec{b}) \cdot (3\vec{a} + 2\vec{b}) \] We can expand this expression using the distributive property of the dot product: \[ (3\vec{a} - 2\vec{b}) \cdot (3\vec{a} + 2\vec{b}) = 9 \vec{a} \cdot \vec{a} + 6 \vec{a} \cdot \vec{b} - 6 \vec{b} \cdot \vec{a} - 4 \vec{b} \cdot \vec{b} \] Since \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \), this simplifies to: \[ 9 \vec{a} \cdot \vec{a} - 4 \vec{b} \cdot \vec{b} \] Now, use the magnitudes of the vectors: \[ \vec{a} \cdot \vec{a} = |\vec{a}|^2 = 9, \quad \vec{b} \cdot \vec{b} = |\vec{b}|^2 = 4 \]
Thus, the expression becomes: \[ 9 \times 9 - 4 \times 4 = 81 - 16 = 65 \]
Thus, the answer is \( 65 \).