Lines and PlanesLinesLet lbe a line parallel to vector v. v=v1i+v2j+v3kLet P0(a, b, c)be a point on the line and P(x, y, z)be any other point on the line.P0Pis a vector parallel to the line.P0Pvso The parametric equations of a line are:x=a+v1ty=b+v2tz=c+v3t
To find the equation of a line you need a point on the line and a vector parallel to the line.ex. Find the equation of the line through P(1, 5, 3) parallel to vector v=2, 0, 1=2i+k.ex. Find the equation of the line through points P(1, 2, 3) and Q(3, 6, 9).ex. Find a vector parallel to the line
x=9-ty=3z=6tex. Determine which of the following line equations represent the same line as x=9-ty=3z=6tx1=7+2tx2=11-3tx3=9+ty1=3y2=3y3=3z1=12-12tz2=10+18tz3=6tPlanes
Let P0(a, b, c)be a point in the plane. If P(x, y, z) is any other point in the plane, then P0Pis a vector that lies in the plane. Let n=n1i+n2j+n3kbe a vector perpendicular (normal) to the plane. Then
The equation of a plane is n1x+n2y+n3z=n⋅pwhere nis the normal vector and pis the vector formed by the point.To find the equation of a plane you need a point on the plane and a vector normal (perpendicular) to the plane.