Calculus
posted by Brandon on .
Find a linear function whose graph is the plane that intersects the xyplane along the line y=2x+2 and contains the point (1,2,2).

a little vector analysis makes this relatively easy. Find two points B and C on the line. Let A=(1,2,2)
Form vectors AB and AC. These two vectors lie in the desired plane. Let n be the normal to the plane.
n = AB × AC
n•(vA) = 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 • (x1)i + (y2)j + (z2)k = 0
2xy+z = 2
For a fuller explanation, see
www.jtaylor1142001.net/calcjat/Solutions/VPlanes/VPPtLine.htm