Question

If \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), then the dot product \( \vec{A} \cdot \vec{B} \) is:

• A. \( 5 \)
• B. \( 6 \)
• C. \( 7 \)
• D. \( 8 \)

Hint:

To calculate the dot product of two vectors, multiply their corresponding components and sum the results.

Answer 1

The Correct Option is A

We are given two vectors \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), and we need to find their dot product. Step 1: Use the formula for the dot product The dot product of two vectors \( \vec{A} = a_1 \hat{i} + a_2 \hat{j} \) and \( \vec{B} = b_1 \hat{i} + b_2 \hat{j} \) is given by: \[ \vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 \] Step 2: Substitute the values of the vectors For \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), we have: \[ \vec{A} \cdot \vec{B} = (2)(4) + (3)(-1) = 8 - 3 = 5 \] Answer: The dot product \( \vec{A} \cdot \vec{B} \) is \( 5 \), so the correct answer is option (1).