Question

Find a linear function whose graph is the plane that intersects the xy-plane along the line y=2x+2 and contains the point (1,2,2).

Answer 1

a little vector analysis makes this relatively easy. Find two points B and C on the line. Let A=(1,2,2)

Form vectors <b>AB</b> and <b>AC</b>. These two vectors lie in the desired plane. Let <b>n</b> be the normal to the plane.

<b>n</b> = <b>AB</b> × <b>AC</b>
<b>n</b>•(<b>v</b>-<b>A</b>) = 0 is the desired plane.

So, pick any two points on the line, say B=(0,2,0) and C=(1,4,0)

AB = -i -2k
AC = 2j -2k

n =
| i j k |
|-1 0 -2 |
| 0 2 -2 |
= 4i -2j + 2k

4i -2j + 2k • (x-1)i + (y-2)j + (z-2)k = 0
2x-y+z = 2

For a fuller explanation, see

www.jtaylor1142001.net/calcjat/Solutions/VPlanes/VPPtLine.htm