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I am having difficulty figuring out this problem. I know that for part a) the values of the variables are the ones that are displayed in the equation of a plane but I am having difficulty with the other two letters. Any help would be greatly appreciated.

What conditions on a, b, c, d will ensure that the plane pi : ax + by + cz = d has each of the following properties?

Write your answers as equations involving only the variables a, b, c, d.

Justify your answers briefly.

(a) pi is the plane 2x -3y = 1.

(b) pi is orthogonal to the plane x + 5y - 9z = 3.

(c) pi is perpendicular to the line L : (x, y, z) = (1, 4, 2)+ < 2, 3,-1 > t.

(a) This means the plane must have c = 0 with 2a - 3b = 1, so a = (1+3b)/2.(a) In order for the

(b) For two planes to be perpendicular, this means that the line of intersection needs to found.

Once this has been done, a vector in the plane that is orthogonal to the line needs to be found.

Once this vector has been found in each plane, the vectors need to be orthogonal to each other.

To be orthogonal to each other, the two vectors need to have a dot product that is 0.

(c) This means the plane has to pass through the point (1,4,2).

It must be of the form 2ax + 3ay - az = 12 for some value a.

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