Question

If \(\vec{a}=2\hat{i}+\hat{j}\), \(\vec{b}=\hat{i}+3\hat{j}\), then \(\vec{a}\cdot\vec{b}=\)

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

Hint:

Dot product: \[ \vec{a}\cdot\vec{b} = a_xb_x+a_yb_y+a_zb_z \] Multiply corresponding components and add.

Answer 1

The Correct Option is A

Concept: Dot product formula: \[ (a_1\hat{i}+a_2\hat{j})\cdot(b_1\hat{i}+b_2\hat{j}) = a_1b_1+a_2b_2 \]

Step 1: Identifying components. \[ \vec{a}=2\hat{i}+\hat{j} \] \[ \vec{b}=\hat{i}+3\hat{j} \] Thus: \[ a_1=2,\quad a_2=1 \] \[ b_1=1,\quad b_2=3 \]

Step 2: Calculating dot product. \[ \vec{a}\cdot\vec{b} = (2)(1)+(1)(3) \] \[ =2+3 \] \[ =5 \] Final Answer: \[ \boxed{(A)\ 5} \]