Question

If $\overrightarrow{ A }=(1,1,1), \overrightarrow{ C }=(0,1,-1)$ are given vectors, then a vector $\overrightarrow{ B }$ satisfying the equations $\overrightarrow{ A } \times \overrightarrow{ B }=\overrightarrow{ C }$ and $\overrightarrow{ A } \cdot \overrightarrow{ B }=3$ is $\ldots$

Answer 1

Let $ \overrightarrow{ B }=x \hat{ i }+y \hat{ j }+z \hat{ k }$
Given, $ \overrightarrow{ A }=\hat{ i }+\hat{ j }+\hat{ k }, \overrightarrow{ C }=\hat{ j }-\hat{ k }$
Also, given $\overrightarrow{ A } \times \overrightarrow{ B }=\overrightarrow{ C }$
$\Rightarrow (z-y) \hat{ i }-(z-x) \hat{ j }+(y-x) \hat{ k }=\hat{ j }-\hat{ k }$
$\Rightarrow z-y=0, x-z=1, y-x=-1$
$\overrightarrow{ A } \cdot \overrightarrow{ B }=3 \Rightarrow x+y+z=3$
On solving above equations, we get
$x=\frac{5}{3}, y=z=\frac{2}{3} $
$\overrightarrow{ B }=\left(\frac{5}{3} \hat{ i }, \frac{2}{3} \hat{ j }, \frac{2}{3} \hat{ k }\right)$