Question

Find an equation of the plane that contains the point p(5,1,4) and the line x = 2−t, y = 3 + t, t = 3−3t.

Answer 1

The vector from the reference point on the line to the point is
.. (5-2, 1-3, 4-3) = (3, -2, 1)
The cross product of this with the line's direction vector (-1, 1, -3) is (5, 8, 1). The dot product of that with any of the points in the problem statement is 37. Your plane equation can be written as
.. 5x +8y +z = 37

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A suitable vector calculator is very handy for this. I used an HP-48 app on my phone.