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

Given vectors \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \), calculate their dot product. Step 1: Recall the dot product formula For vectors \( \vec{A} = a_1 \hat{i} + a_2 \hat{j} \) and \( \vec{B} = b_1 \hat{i} + b_2 \hat{j} \), the dot product is defined as: \[ \vec{A} \cdot \vec{B} = a_1 b_1 + a_2 b_2 \] Step 2: Apply the formula with given vector components Substitute the components of \( \vec{A} = 2\hat{i} + 3\hat{j} \) and \( \vec{B} = 4\hat{i} - \hat{j} \) into the formula: \[ \vec{A} \cdot \vec{B} = (2)(4) + (3)(-1) = 8 - 3 = 5 \] Answer: The calculated dot product \( \vec{A} \cdot \vec{B} \) is \( 5 \).