Question

If $\overline{a}=(1,1,1);\overline{c}=(0,1,-1)$ are two given vectors, find $\overline{b}$ such that $\overline{a}\times \overline{b}=\overline{c}$ and $\overline{a}.\overline{b}=3$

Answer 1

Let $\overrightarrow{b}=x\hat{i}+y\hat{j}+z\hat{k}=(x,y,z)$

Given that

$\Rightarrow \overrightarrow{a}.\overrightarrow{b}=3$

$\Rightarrow (1,1,1,).(x,y,z)=3$

$\Rightarrow x+y+z=3$ ...$\left(1\right)$

Given that

$\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$

$\Rightarrow (1,1,1,)\times (x,y,z)=(0,1,-1)$

$\Rightarrow (z-y,x-z,y-x)=(0,1,-1)$

$\Rightarrow y=z$

$\Rightarrow y-x=-1\Rightarrow x=y+1$

Substitute these values in $\left(1\right)$, we get

$\Rightarrow 1+y+y+y=3$

$\Rightarrow y=\frac{2}{3}$

$\Rightarrow z=\frac{2}{3};x=\frac{5}{3}$