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 \( 5 \).