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)

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=03x-y+2z-2=07x-7y-14=0

let, y = t then x = t+2Sub 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

(-) -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+113

11x+y+13z+45

x+44y+13z+53

(n1 X n2) *n3 not = 0

9.5 + 9.6 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

9.5 + 9.6 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