Question

If \(\overset{̄}{a} = 4\hat{i}+\hat{j}+\hat{k}\), \(\overset{̄}{b} = 2\hat{i}+\hat{j}+2\hat{k}\) and \(\overset{̄}{c} = 3\hat{i}+4\hat{j}+5\hat{k}\), then \((\overset{̄}{a}+\overset{̄}{b})\cdot (\overset{̄}{b}+\overset{̄}{c}) =\)

• A. \(30\)
• B. \(21\)
• C. \(61\)
• D. \(10\)

Hint:

Add the vectors first, then take the dot product.

Answer 1

The Correct Option is C

Step 1: Expand algebraically
$(\vec a+\vec b)\cdot(\vec b+\vec c) = \vec a\cdot\vec b+\vec a\cdot\vec c+|\vec b|^2+\vec b\cdot\vec c$.

Step 2: Evaluate each
$\vec a\cdot\vec b = 8+1+2 = 11$, $\vec a\cdot\vec c = 12+4+5 = 21$, $|\vec b|^2 = 9$, $\vec b\cdot\vec c = 6+4+10 = 20$.

Step 3: Sum
$11+21+9+20 = 61$. Option (C).

Final Answer:
61. \[ \boxed{\text{(C)}\ 61} \]