Find the vector and cartesian equations of the plane which passes through the point `(5, 2, 4)`and perpendicular to the line with direction ratios `(2, 3, 1)`.

Explanation

Let `A(5,2,4)` is the point passing through given plane.

In vector form,

`vecA = 5hati+2hatj+4hatk`

Let `vecr` is another vector on that plane, then, line joining `vecr` and `veca` will be,

`vecr - vecA`

We are given another line with direction cosines (2,3,1) that is perpendicular to the first line.

So, dot product of these two lines will be `0`.

`:. (vecr - (5hati+2hatj+4hatk)).(2hati+3hatj+hatk) = 0`

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