Question

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\overrightarrow{a}+(\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{0}$ and $\overrightarrow{b}\cdot \overrightarrow{c}=5$. Then, the value of $3(\vec{c}\cdot \vec{a})$ is equal to ______

Answer 1

Answer: 0

$\vec{a}+\vec{b}\times \vec{c}=0$
$\vec{a}\times \vec{b}+|\vec{b}{|}^2\vec{c}-5\vec{b}=0$
It gives $\vec{c}=\frac{1}{3}(10\hat{i}+3\hat{j}+2\hat{k})$
so $3\vec{a}.\vec{c}=10$
But it does not satisfy $\vec{a}+\vec{b}\times \vec{c}=0$.
This question has data error.
Alternate (Explanation) :
According to given $a$ & $b$
$a\cdot b=1-2+3=2\dots$ (i)
but given equation
$\vec{a}=-(\vec{b}\times \vec{c})$
$\Rightarrow \vec{a}\perp \vec{b}\Rightarrow \vec{a}\cdot \vec{b}=0$
which contradicts.