Step 1: Understanding the Concept:
The dot product, also known as the scalar product, is an operation on two vectors that results in a scalar (a single number).
It measures how much one vector "projects" onto another.
In component form, the dot product is calculated by multiplying corresponding components of the vectors and summing the products.
Step 2: Key Formula or Approach:
For two vectors \( \vec{u} = u_1\hat{i} + u_2\hat{j} \) and \( \vec{v} = v_1\hat{i} + v_2\hat{j} \):
\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 \]
Step 3: Detailed Explanation:
The given vectors are:
\[ \vec{a} = 2\hat{i} + 1\hat{j} \]
\[ \vec{b} = 1\hat{i} + 3\hat{j} \]
Identify the individual components:
For \( \vec{a} \): \( a_x = 2, a_y = 1 \).
For \( \vec{b} \): \( b_x = 1, b_y = 3 \).
Now, perform the dot product calculation:
\[ \vec{a} \cdot \vec{b} = (a_x \cdot b_x) + (a_y \cdot b_y) \]
Substitute the component values:
\[ \vec{a} \cdot \vec{b} = (2 \cdot 1) + (1 \cdot 3) \]
\[ \vec{a} \cdot \vec{b} = 2 + 3 \]
\[ \vec{a} \cdot \vec{b} = 5 \]
Step 4: Final Answer:
The scalar dot product of the two vectors is \( 5 \).
This matches Option (A).