Do you know that there exist a unique plane given two intersecting lines? If you are given pointr0 and vectorv, thenr0+ s[/b]v[/b] gives all points on the line throughr0 in the direction of vectorv.r0+ tvgives all points on the line throughr0 in the direction ofw. Those two lines, given by t= 0 and s= 0 respectively, determine the plane. Given specific values of s and v, "r0+ sv+ tw" means "starting at ro, go for a distance s in the direction of vector v, the go distance t in the direction of vector w" to get to the corresponding point on the plane.

In order that, forra point in the plane, (r-r0).n= 0,nmust beperpendicular(ornormal) to the plane. And that is given by the cross product of two vectors in the plane.

In your example, (9, -5, 9) and (3, -2, 3) are vectors in the plane. Their cross product,n= (3, 0, -3), is perpendicular to the plane. For any point in the plane, (x, y, z), we must have ((x, y, z)- (-3, 4, -9)).(3, 0, -3)= 0 or (x+ 3)(3)+ (y+ 4)(0)+ (z+ 9)(-3)= 3x+ 9- 3z- 27= 3x- 3z- 18= 0.

Now, you were given that x= -3+ 9s+ 3t, y= 4- 5s- 2t, z= -9+ 9s+ 3t. Putting those into 3x- 3z- 18, we have 3(-3+ 9s+ 3t)+ 0(4- 5x- 2t)- 3(-9+ 9s+ 3t)- 18= (-9+ 27s+ 9t)+ (27- 27s- 9t)- 18= (-9+ 27- 18)+ (27x- 27s)+ (9t- 9t)= 0 so it does, in fact, satisfy the equation.