Question

If $\vec{a} \cdot \vec{b}=12$, then $(3\vec{a}) \cdot (3\vec{b})$ is equal to:

• A. 36
• B. 4
• C. 108
• D. 16
• E. 144

Hint:

Multiply all scalars together before multiplying with the dot product result.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to evaluate a dot product involving scalar multiples of vectors. This requires knowing the properties of scalar multiplication in relation to the dot product.
Step 2: Key Formula or Approach:
The key property of the scalar (dot) product is that scalars can be factored out:
For any scalars k and l, and vectors \( \vec{u} \) and \( \vec{v} \):
\[ (k\vec{u}) \cdot (l\vec{v}) = (kl)(\vec{u} \cdot \vec{v}) \] Step 3: Detailed Explanation:
We are asked to compute \( (3\vec{a}) \cdot (3\vec{b}) \).
Using the property from Step 2, where \( k=3 \), \( l=3 \), \( \vec{u}=\vec{a} \), and \( \vec{v}=\vec{b} \):
\[ (3\vec{a}) \cdot (3\vec{b}) = (3 \times 3)(\vec{a} \cdot \vec{b}) \] \[ = 9(\vec{a} \cdot \vec{b}) \] We are given in the problem that \( \vec{a} \cdot \vec{b} = 12 \).
Substitute this value into our expression:
\[ 9 \times 12 = 108 \] Step 4: Final Answer:
The value of \( (3\vec{a}) \cdot (3\vec{b}) \) is 108.