Find the angle between the lines ` vec r= dot( hat i+2 hat j- hat k)+lambda( hat i- hat j+ hat k)` and the plane ` vec r . dot(3 hat i- hat j+ hat k)=4.`

Explanation

Now acute angle `theta` between a line `vec{r}=vec{a}+lambda vec{b}` and plane `vec{r} cdot vec{n}=d` is

`sin theta=frac{vec{b} cdot vec{n}}{|vec{b} cdot vec{n}|}`

So here,

`sin theta=frac{2-1+1}{sqrt{3} times sqrt{6}}`

`sin theta=frac{2}{sqrt{18}}`

`theta=sin ^{-1}(frac{2}{sqrt{18}})`

`therefore` angle between the line a vector perpendicular to the plane is `sin ^{-1}(frac{sqrt{2}}{3})`.

`therefore` the angle between plane and line is `frac{pi}{2}-sin ^{-1}(frac{sqrt{2}}{3})=cos ^{-1}(frac{sqrt{2}}{3})`