Question

If \( \mathbf{a} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} \), find \( \mathbf{a} \cdot \mathbf{b} \) (the dot product).

• A. 2
• B. 4
• C. 10
• D. 12

Hint:

Remember: The dot product of two vectors is calculated by multiplying their corresponding components and summing them.

Answer 1

The Correct Option is A

Given vectors:

\( \mathbf{a} = 3\hat{i} + 4\hat{j} \)

\( \mathbf{b} = 2\hat{i} - \hat{j} \)

Step 1: Dot Product Formula

The dot product of vectors \( \mathbf{a} = a_1 \hat{i} + a_2 \hat{j} \) and \( \mathbf{b} = b_1 \hat{i} + b_2 \hat{j} \) is \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).

Step 2: Apply Formula

For \( \mathbf{a} = 3\hat{i} + 4\hat{j} \) and \( \mathbf{b} = 2\hat{i} - \hat{j} \), the components are \( a_1 = 3, \, a_2 = 4, \, b_1 = 2, \, b_2 = -1 \). Substituting these into the formula:

\[ \mathbf{a} \cdot \mathbf{b} = (3)(2) + (4)(-1) = 6 - 4 = 2 \]

Result:

The dot product \( \mathbf{a} \cdot \mathbf{b} \) equals \( 2 \).