Lines and PlanesLinesLet lbe a line parallel to vector v. vv1iv2jv3kLet P0(a, b, c)be a point on the line and P(x, y, z)be anyother point on the line.P0Pis a vector parallel to the line.P0PvsoThe parametric equations of a line are:xav1tybv2tzcv3t
To find the equation of a line you need a point on the line and avector parallel to the line.ex. Find the equation of the line through P(1, 5, 3) parallel tovector v2, 0, 12ikex. Find the equation of the line through points P(1, 2, 3) andQ(3, 6, 9)..
ex. Find a vector parallel to the linex9-ty3z6tex. Determine which of the following line equations represent thesame line as x9-ty3 z6tx172tx211-3tx39ty13 y23 y33z112-12tz21018tz36t
PlanesLet P0(a, b, c)be a point in the plane. If P(x, y, z) is any otherpoint in the plane, then P0Pis a vector that lies in the plane. Letnn1in2jn3kbe a vector perpendicular (normal) to theplane. Then
vector normal (perpendicular) to the plane.ex. Find the equation of the plane through P(2, 6, 3) which isnormal to n3, -2, 1ex. Find the equation of the plane through P(2, 6, 3), Q(-1, 0, 1)and R(4, 4, 4).n.
ex. Find a normal vector to the plane z2xy1ex. Determine if the following two lines are parallel,perpendicular, intersect at a non right angle, or are skew..x135tx22sy11-5ty2-7-sz110tz23-2s
Do: 1. Find the equation of the line through P(1,1,1) perpendicularto the plane 2x-3y5z42. Find the equation of the plane through P(2, 5, 1) parallel to theplane z = -3.3. Find the equation of the line of intersection of planesx + y + z = 1 and x – 2z = 0.4. Is the line x-12t,y-2t,zthe same line asx212t,y2t,z?.2