Find the equation of the plane through the line of intersection of ` vec rdot(2 hat i-3 hat j+4 hat k)=1a n d vec rdot( hat i- hat j)+4=0` and passing through `(2,-1,1)`

Explanation

Equation of Plane 1 is ` vec{r}(2hat{i}-3hat{j}+4hat{k})=1 `

Equation of Plane 2 is `vec{r}(hat{i}- hat{j})+4=0`

`[vec{r}(2hat{i}-3hat{j}+4hat{k})-1]+lambda[vec{r}( hat{i}- hat{j})+4]=0`

`[(2hat{i}-1hat{j}+hat{k}) cdot(2hat{i}-3hat{j}+4hat{k})-1]+lambda[(2hat{i}-hat{j}+hat{k}) cdot(hat{i}-hat{j})+4]=0`

`implies(4+3+4-1)+lambda(2+1+4)=0`

`Rightarrow 10+7lambda=0`

`implies lambda=frac{-10}{7}`

`vec{r}[(2hat{i}-3hat{j}+4hat{k})+lambda( hat{i}- hat{j})]=1-4 lambda`

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