Question

If \(|\overset{̄}{a}| = 4,|\overset{̄}{b}| = 3\) and \(\overset{̄}{a}\cdot \overset{̄}{b} = 8\) then \([\overset{̄}{a} \overset{̄}{a}+\overset{̄}{b} \overset{̄}{a}\times \overset{̄}{b}] =\)

• A. \(96\)
• B. \(80\)
• C. \(64\)
• D. \(120\)

Hint:

Expand the triple product and use the vector triple product identity.

Answer 1

The Correct Option is B

Step 1: Expand the triple product:
\([\vec a,\ \vec a+\vec b,\ \vec a\times\vec b]=\vec a\cdot[(\vec a+\vec b)\times(\vec a\times\vec b)]\).

Step 2: Split the cross product:
\((\vec a+\vec b)\times(\vec a\times\vec b)=\vec a\times(\vec a\times\vec b)+\vec b\times(\vec a\times\vec b)\). The first term is perpendicular to \(\vec a\), so its dot with \(\vec a\) is 0.

Step 3: Use the triple product identity:
\(\vec b\times(\vec a\times\vec b)=\vec a(\vec b\cdot\vec b)-\vec b(\vec b\cdot\vec a)\). So \(\vec a\cdot[\dots]=|\vec a|^2|\vec b|^2-(\vec a\cdot\vec b)^2\).

Step 4: Substitute:
\(=16\cdot9-8^2=144-64=80\). Option B.

Final Answer:
The value is 80. \[ \boxed{\text{(B) }80} \]