Mathematics - Vector Question with Solution | TestHub

Given $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\vec{j}-\vec{k}$,
Let $\bar{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$
Now, $\hat{j}-\hat{k}=\bar{a}\times \bar{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ b_2& b_2& b_3\end{array}\right|$
$\Rightarrow b_3-b_2=0,b_1-b_3=1,b_2-b_1=-1$
$\Rightarrow b_3=b_2,b_1=b_2+1$
Also, $\bar{a}\cdot \bar{b}=1$
$\Rightarrow b_1+b_2+b_3=1$
$\Rightarrow 3b_2+1=1$
$\Rightarrow b_2=0$
$\Rightarrow b_1=1,b_3=0$
Thus, $\bar{b}=\hat{i}$