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I am having difficulty figuring out this oproblem. 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) For this, this can just read off the coefficients for the scalar eqn for pi: a = 2, b = -3, c = 0, d = 1.

b) If planes are othogonal then their normals are orthogonal: <1,5,-9>･<a,b,c> = 0 ⇒ a + 5b -9c = 0; also d = -3.

c) The normal to pi is parallel to the perpendicular line so that a = 2, b = 3, c = -1.

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