Question

Given \(\quad \vec{A}=(2 \hat{i}-3 \hat{j}+\hat{k}), \quad \vec{B}=(3 \hat{i}+\hat{j}-2 \hat{k})\) and \(\vec{C}=(3 \hat{i}+2 \hat{j}+\hat{k}) \cdot(\vec{A}+\vec{B}) \cdot \vec{C}\) will be

• A. 10
• B. 12
• C. 18
• D. 20

Hint:

Add the first two vectors component-wise, then take the dot product with the third.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The sum is found by adding corresponding components, and the dot product multiplies matching components and adds them.

Step 2: Add the vectors
\(\vec A+\vec B=(2+3)\hat i+(-3+1)\hat j+(1-2)\hat k=5\hat i-2\hat j-\hat k\)

Step 3: Calculation
\((\vec A+\vec B)\cdot\vec C=(5)(3)+(-2)(2)+(-1)(1)=15-4-1=10\)

Step 4: Compare the options
The value 10 matches option (A).

Final Answer:
The required dot product equals 10. \[ \boxed{\text{(A) }10} \]