Question
Three vectors $\vec a,\vec b,\vec c$ are such that $|\vec a|=2,\ |\vec b|=2,\ |\vec c|=4$ and $\vec a+\vec b+\vec c=0$. Then the value of $4\vec a\cdot\vec b+3\vec b\cdot\vec c+3\vec c\cdot\vec a$ will be
This question was previously asked in
WBJEE 2026 Physics and Chemistry Question Paper (24-May-2026)
The correct answer is
$-32$
Vector Dot Product Calculation Strategy
We are given three vectors $\vec a$, $\vec b$, $\vec c$ with magnitudes $|\vec a|=2$, $|\vec b|=2$, $|\vec c|=4$, and the condition $\vec a+\vec b+\vec c=0$. We need to find the value of the expression $4\vec a\cdot\vec b+3\vec b\cdot\vec c+3\vec c\cdot\vec a$.
Deriving Dot Products from Vector Sum
From the given condition $\vec a+\vec b+\vec c=0$, we can derive individual dot products:
- Since $\vec a+\vec b = -\vec c$, squaring both sides gives $(\vec a+\vec b)\cdot(\vec a+\vec b) = (-\vec c)\cdot(-\vec c)$. This expands to $|\vec a|^2 + |\vec b|^2 + 2\vec a\cdot\vec b = |\vec c|^2$. Substituting the magnitudes: $2^2 + 2^2 + 2\vec a\cdot\vec b = 4^2$, which simplifies to $4 + 4 + 2\vec a\cdot\vec b = 16$. Solving for $\vec a\cdot\vec b$: $2\vec a\cdot\vec b = 16 - 8 = 8$, so $\vec a\cdot\vec b = 4$.
- Similarly, since $\vec b+\vec c = -\vec a$, squaring gives $|\vec b|^2 + |\vec c|^2 + 2\vec b\cdot\vec c = |\vec a|^2$. Substituting magnitudes: $2^2 + 4^2 + 2\vec b\cdot\vec c = 2^2$, which means $4 + 16 + 2\vec b\cdot\vec c = 4$. Solving for $\vec b\cdot\vec c$: $2\vec b\cdot\vec c = 4 - 20 = -16$, so $\vec b\cdot\vec c = -8$.
- Likewise, since $\vec c+\vec a = -\vec b$, squaring gives $|\vec c|^2 + |\vec a|^2 + 2\vec c\cdot\vec a = |\vec b|^2$. Substituting magnitudes: $4^2 + 2^2 + 2\vec c\cdot\vec a = 2^2$, which is $16 + 4 + 2\vec c\cdot\vec a = 4$. Solving for $\vec c\cdot\vec a$: $2\vec c\cdot\vec a = 4 - 20 = -16$, so $\vec c\cdot\vec a = -8$.
Calculating the Final Expression
Now substitute the calculated dot product values into the target expression $4\vec a\cdot\vec b+3\vec b\cdot\vec c+3\vec c\cdot\vec a$:
Expression $= 4(\vec a\cdot\vec b) + 3(\vec b\cdot\vec c) + 3(\vec c\cdot\vec a)$
Expression $= 4(4) + 3(-8) + 3(-8)$
Expression $= 16 - 24 - 24$
Expression $= 16 - 48$
Expression $= -32$.
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