X/ 3/ ' Example # 1: Determine the point(s) of intersection of each line and plane, if possible. a) L1: [x,y,z] = [4,2,6] +t[1,-2,3] and .71'12 13x+7y2216=0 b) L1: x;4=%=z+11 and 7:1:3x2y+4z8=0 0) L1: x=5+3t and lz4x5y-4z+2=0 y=-2+4t z=92t Solution: */ Example # 1: Write vector and parametric equations for the plane which contains the points A (1,2,-3) , B (5,1,0) and C (3,2,6). Solution: Cartesian Equation of a Plane (also known as Scalar Equation of a Plane) This is an equation without using parameters. To determine the scalar equation of a plane, we need to nd the normal vector for the plane (ie: a vector perpendicular to the plane). This can be done by taking the cross product of two non-collinear vectors in the plane. The result is Let A (a1 , a2 , 213) be a xed point on a line in R3 with direction vector m = [m1 , m2 , m3]. Let P (x,y,z) be any point on the line. The equations of the line can be written in the following forms; Vector Equation: Parametric Equation: Symmetric Equation