Question

What vector when added to $(2\hat{i}-2\hat{j}+\hat{k})$ and $(2\hat{i}-\hat{k})$ will give a unit vector along negative $y-$axis?

Answer 1

$(2\hat{i}-2\hat{j}+\hat{k})+(2\hat{i}-\hat{k})=(4\hat{i}-2\hat{j}+0\hat{k})$

Now we have to find a vector which when added to the above vector gives along negative y-axis i.e. $-\hat{j}$.

Let the required vector be $x\hat{i}+y\hat{j}+z\hat{k}$

$\begin{array}{c}\Rightarrow (4\hat{i}-2\hat{j}+0\hat{k})+(x\hat{i}+y\hat{j}+z\hat{k})=-\hat{j}\\ \Rightarrow (4+x)\hat{i}+(-2+y)\hat{j}+(0+z)\hat{k}\\ 4+x=0\Rightarrow x=-4\\ -2+y=-1\Rightarrow y=1\\ 0+z=0\Rightarrow z=0\end{array}$

Therefore the required vector is $-4\hat{i}+\hat{j}$