1.

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)`.

Answer» 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`
`=>vecr.(2hati+3hatj+hatk) - 10-6-4 = 0`
`=>vecr.(2hati+3hatj+hatk) = 20`
We can take, `vecr = ahati+bhatj+chatk `
Then, our equation becomes,
`2a+3b+c = 20`, which is required vector equation.
Cartesian equation will be,
`2x+3y+z = 20`


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