9.1 INTERSECTION OF TWO LINES To find the intersection of two lines L(1) and L(2) :  Write both equations in parametric form. L(1)= (x1,y1,z1) + t(a,b,c) L(2)=(x2,y2,z2) + s(a,b,c)  Now, solve for s and t algebraically. x1 + t(a) = x2 + s(a) y1 + t(b) = y2 + s(b) z1 + t(c) = z2 + s(c)  Substitute for either s or t to find the P.O.I

9.1 INTERSECTION OF TWO LINES For the above system of equations, you can have three types of solutions:- UNIQUE SOLUTION: You get one value each for s and t. INFINITE SOLUTION: The system of equations has infinite solution. 0(t) = 0 NO SOLUTION: The system of equations does not have a solution. 0(t) = 7

9.1 INTERSECTION OF A LINE AND A PLANE To find the intersection in between a plane and a line Write equation of line in parametric form. Then plug into the Cartesian equation of the plane. Question: find the point of intersection between Line L1 = (-6,-9,-1) + t(-2,3,1) and Plane P1 = -x+2y+z+4 t = -3 POI = (0,0,-4)

9.2 SYSTEM OF EQUATIONS For a linear system of equations you can have ___,___ or ____number of solutions Question: 2x + y = -9 x + 2y = -6 A system of equation is _________ if it has one or infinite solutions A system is ______________ if it has no solutions 0,1 infinite x= -4 y= -1 consistent, inconsistent

9.3 intersection of two planes Two planes can intersect in three ways:-

9.3 intersection of two planes INTERSECTION: The solution is finite. Both the planes intersect on a line. Example: find the line of intersection b/w -2x+3y+z+6=0 and 3x-y+2z-2=0 -2x+3y+z+6=0 multiply by (2) 3x-y+2z-2=0 -4x+6y+2z+12=0 3x-y+2z-2=0 7x-7y-14=0 let, y = t then x = t+2 Sub values into any plane equation, you get z = -t-2 Line of intersection is L1 = (2,0,-2) + t(1,1,-1)

9.3 intersection of two planes COINCIDENT: the result for this set of equations is infinite. Example: find the intersection b/w -2x+3y+z+6=0 and -4x+6y+2z+12=0 -2x+3y+z+6=0 multiply by (2) -4x+6y+2z+12=0 (-) -4x+6y+2z+12=0 0

9.3 intersection of two planes PARALLEL: the result for this set is not possible. Example: find the intersection b/w -2x+3y+z+6=0 and -4x+6y+2z+1=0 -2x+3y+z+6=0 multiply by (-2) -4x+6y+2z+1=0 4x-6y-2z-12=0 -4x+6y+2z+1=0 -11

9.4 intersection of three planes 2x+3y+4z+2 4x+6y+8z+5 6x+9y+12z+99 n1 = n2 = n3

9.4 intersection of three planes 2x+3y+4z+2 4x+6y+8z+4 6x+9y+12z+6 n1 = n2 = n3 D1 = D2 = D3

9.4 intersection of three planes 2x+3y+4z+99 4x+6y+8z+4 6x+9y+12z+6 n1 = n2 = n3 D1 = D2

9.4 intersection of three planes 2x+3y+4z+99 4x+6y+8z+4 3x+5y+z+57 n1 = n2

9.4 intersection of three planes 3x+5y+7z+99 4x+6y+8z+4 6x+9y+12z+6 n1 = n2 D1 = D2

9.4 intersection of three planes 3x+5y+7z+99 11x+y+13z+4 x+44y+13z+5 (n1 X n2) *n3 = 0

9.4 intersection of three planes 3x+5y+7z+99 11x+y+13z+4 x+44y+13z+5 (n1 X n2) *n3 = 0

9.4 intersection of three planes 3x+5y+7z x+y+13z+45 x+44y+13z+53 (n1 X n2) *n3 not = 0

DISTANCE To calculate distance of a point from a line we have two formulas When the point and line are in 2-D When the point and line are in 3-D

DISTANCE To calculate distance of a point from a plane we have two formulas When the point and plane are in 2-D When the point and plane are in 3-D

THINKING QUESTIONS Solve using matrices: x - 5y + 2z = 27 3x + 2y - z = -5 4x - 3y + 5z = 42 (2,-3,5)

THINKING QUESTIONS Find parametric equations of the plane through the points A(2,-1,1), B(4,1,5), C(1,2,2). Find the value of k if point (0, k, -3) is in the plane (1,2,2)+s(1,1,2)+t(-1,3,1) k=-3