Find the vector equation of the following planes in non parametric form: ` -> r=(2 hat i+2 hat j- hat k)+lambda( hat i+2 hat j+3 hat k)+mu(5 hat i-2 hat j+7 hat k)dot`

Explanation

Given: `vec{r}=(3 hat{j})+lambda(hat{i}+2 hat{k})+mu(-2 hat{1}-hat{j}+hat{k})`

It is known that `r=vec{a}+lambda vec{b}+mu c` represents a plane passing through a point having position vector `vec{a}` and parallel to the vectors `vec{b}` and `vec{c}`

Clearly,:

`vec{a}=3 hat{j}, vec{b}=hat{i}+2 hat{k}` and `c=-2 hat{1}-hat{j}+hat{k}`

Now, the plane is perpendicular to `vec{b} times c`

Therfore, `vec{n}=vec{b} times vec{c}=|[hat{i}, hat{j}, hat{k}], [1 ,0 , 2],[ -2, -1, 1] |`


`=(0+2) hat{i}-(1+4) hat{j}+(-1+0) hat{k}`

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