UNIT 4 - RELATIONSHIPS BETWEEN LINES AND PLANES Date ... · UNIT 4 - RELATIONSHIPS BETWEEN LINES AND PLANES Date Lesson § TOPIC Homework Mar. 23 4.1 (26) 9.1 The Intersection of

UNIT 4 - RELATIONSHIPS BETWEEN LINES AND PLANES

Date Lesson § TOPIC Homework

Mar. 23 4.1

(26) 9.1

The Intersection of a Line with a Plane and the

Intersection of Two Lines

Pg. 496 # (4, 5)b, 7, 8b, 9bd, 12

Mar. 27 4.2

(27) 9.2

Systems of Equations Pg. 507 # (3, 5, 6)a, 9, 12

Mar. 28 4.3

(28) 9.3

Intersection of Two Planes Pg. 516 # 1a, 2a, 5, 6, 10

Mar. 29 4.4

(29) 9.4

Intersection of Three Planes Pg. 530 # 1, 2, 4, 8ad, 9bc, 13af

I.S. 9.5

9.6

Distance from Point to Line & Point to Plane Pg. 540 # (1, 2, 3, 5)a, 6b, 9

Pg. 550 # 1, 2ace, 3, 5

Mar.

30/31

4.5

(30)

Review for Unit 4 Test Pg. 552 # 1-3, 6, 8, 10, 12, 14, 19,

21

Apr. 3 4.6

(31)

TEST - UNIT 4

MCV 4U Lesson 4.1 The Intersection of a Line and a Plane and The Intersection of Two Lines

INTERSECTION of a LINE and a PLANE

Zero solutions One solution Infinite # of solutions

Note: If a line is parallel to the plane, then the dot product of a normal to the plane and the direction vector

of the line will be zero.

Methods to solve a system:

Write each equation in parametric form and solve for each variable.

OR

If a scalar equation is given, directly substitute each component into the equation.

Ex. 1 Determine the intersection of: 02454 zyx ----

tz

ty

tx

1

34

25

----

Ex. 2 Determine the intersection of: 087193 zyx ----

5

3

2

1

1

2

zyx ----

Ex. 3 Determine, without solving, if each line intersects the plane.

a) l1: r (2,5,3) s(3,2,1) b) l2: r (1,0,1) t(2,1,4)

1 : 3x y z 6 2 : 4x 2z 11

INTERSECTION of TWO LINES

Recall: A linear equation can be written in the form a1x1 a2 x2 a3x3 ....... anxn k ,

where a1, ... and k are constants and x1, .... are variables.

A system of linear equations may have: a) Exactly one solution

b) No Solutions

c) An infinite number of solutions

In 3–space, lines that that are neither intersecting nor parallel are said to be "skew".

If a system has at least one solution, the system is said to be "consistent".

A system is "inconsistent" if it has no solutions.

Methods to solve a system of equations include substitution and elimination.

Ex. 4 Find the intersection of the lines.

a)

sy

sxr

ty

txr

51

712

42

3121 b)

)5,5,1()13,9,3(

)3,1,2()6,2,7(

2

1

tr

sr

Pg. 496 # (4, 5)b, 7, 8b, 9bd, 12

MCV 4U Lesson 4.2 Systems of Equations

Number of Solutions to a Linear System of Equations

A linear system of equations can have zero, one or an infinite number of solutions.

Definition of Equivalent Systems

Two systems of equations are defined as equivalent if every solution to one system is also a solution to the

second system of equations

Elementary Operations Used to Create Equivalent Systems of Equations

1. Multiply an equation by a nonzero constant.

2. Interchange any pair of equations.

3. Add a nonzero multiple of one of one equation to a second equation to replace the second equation

Consistent and Inconsistent Systems of Equations

A system of equations is consistent if it has either one solution or an infinite number of solutions.

A system of equations is inconsistent if it has no solution.

Ex. 1 Solve.

19725 zyx ---

8 zyx ---

143 zyx ---

Ex. 3 Determine the value of k for which the system below has :

12 yx ---

kykx 22 ---

a) no solutions

b) 1 solution

c) infinite solutions

Pg. 507 # (3, 5, 6)a, 9, 12

MCV 4U Lesson 4.3 Intersection of Two Planes

Given two planes in three–space, there are three possible geometric models for the intersection of the

planes. If the planes are parallel and distinct, they do not intersect and there is no solution. If the planes

are coincident, every point on the plane is a solution. If two distinct planes intersect, the solution is the set

of points that lie on the line of intersection.

Ex. 1 Describe how the planes in each pair intersect and if they intersect, find the solution.

a)

1 : 2x y z 1 0

2 : x y z 6 0

When determining if planes are

parallel for the purpose of

determining the intersection of

planes it is useful to include the

constant to determine if parallel

planes are distinct or coincident.

Write normals in simplest form.

Write as (A, B, C); D

b)

3 : 2x 6 y 4z 7 0

4 : 3x 9 y 6z 2 0 c)

5 : x y 2z 2 0

6 : 2x 2y 4z 4 0

Ex. 2 Describe how the planes intersect.

r1 (2,4,9) s(3,1,4) t(2,1,1)

r2 (4,8,1) m(4,2,1) n(3,1,1)

Pg. 516 # 1a, 2a, 5, 6, 10

MCV 4U Lesson 4.4 Intersection of Three Planes

A system of three planes is consistent if it has one or more solutions.

The planes intersect at a point. The planes intersect in a line. The planes are coincident.

There is exactly one solution. There are an infinite number of There are an infinite number

The normals are NOT parallel and not solutions. The normals are coplanar, of solutions. The normals are

coplanar. but not parallel. parallel and coplanar.

A system of three planes is inconsistent if it has no solution.

The three planes are parallel and Two planes are parallel and distinct The planes intersect in pairs.

at least two are distinct. The distinct. The third plane is not Pairs of planes intersect in

normals are parallel. parallel. Two of the normals are lines that are parallel and

parallel. distinct. The normals are

coplanar but not parallel.

It is easy to check if normals are parallel; each one is a scalar multiple of the others.

To check if normals are coplanar, use the triple scalar product n1 (n2 n3) .

Remember that this product gives us the volume of a parallelepiped defined by the three vectors. If the

product is zero, the volume is zero and the vectors must be coplanar. If the product is not zero the vectors

are not coplanar.

Ex. 1 Determine the intersection for each set of planes.

a) b)

1 : 2x y 6z 7 0

2 : 3x 4y 3z 8 0

3 : x 2y 4z 9 0

4 : x 5y 2z 10 0

5 : x 7 y 2z 6 0

6 : 8x 5y z 20 0

Ex. 2 Determine if each system can be solved; then solve the system or describe it.

a) 3x + y – 2z = 12 b) x + 3y – z = –10 c) 4x – 2y + 6z = 35

x – 5y + z = 8 2x + y + z = 8 –10x + 5y – 15z = 20

12x + 4y – 8z = –4 x – 2y + 2z = –4 6x – 3y + 9z = –50

One Point of IntersectionOne Point of Intersection

Source: www.jbrookman.me.uk/graphics/index.html

One Line of IntersectionOne Line of Intersection

Triangular Triangular Prism Prism ––

NoNo IntersectionIntersection

Two Parallel Planes Two Parallel Planes ––

No IntersectionNo Intersection

Three Parallel Planes Three Parallel Planes ––

No IntersectionNo Intersection

Pg. 530 # 1, 2, 4, 8ad, 9bc, 13af

MCV 4U Independent Study 4.5 Distance from a Point to a Line in R2 and R3

DISTANCE from a POINT to a LINE in R2

To find the distance from a point Q(x1, y1 ) to a line with scalar equation Ax By C 0 , we can

let a point on the line be P( xo, yo ) and the distance be d .

n Q

R

P

The distance from a point (x1, y1 ) to the line Ax By C 0 is given by the formula

d =

Ax1 By1 C

A2 B2

Ex. 1 Find the distance between the lines

2x – 3y + 12 = 0 and 2x + 3y – 15 = 0.

)3,2()3,2( 21 nn

parallel - need only find a point on one line and find distance from that point to the other line.

For )4,0(,01232 lineonpointyx

find distance from (0, 4) to 2x + 3y – 15 = 0

83.0

13

3

)3()2(

15)4(3)0(2

22

22

11

BA

CByAxd

d = PQ = ntoonPQproj

)(

n

nPQ

=

( x1 xo , y1 yo) ( A,B)

A2 B2

=

Ax1 Axo By1 Byo

A2 B2,

(xo, yo ) is a point on the line Axo and Byo

are constants. Axo Byo C

So, d =

Ax1 By1 C

A2 B2

DISTANCE from a POINT to a LINE in R3

Rsmsrr o ,

P

d d = distance b/w P and the line

T P = a point that is known

Q = any point on the line whose coords are known

R T = point on line such that QT is a vector

representing the direction m

, which is known

m

Q

In PQR, sin = QP

d

d = sinQP from cross product, we know sinQPmQPm

If we substitute d = sinQP into this formula , we find that

)(dmQPm

Solving for d gives d = m

QPm

The distance from a point P ),,( 111 zyx to the line Rsmsrr o ,

, in R3, is given by the formula

d = m

QPm

, where Q is a point on the line and P is any other point, both of whose coordinates

are known, and m

is the direction vector of the line.

Ex. 2 Find the distance between the point A(2, -3, 5) and the line Rssr ),3,1,4()2,5,0(

)3,8,2()5,3,2()2,5,0( QPPQ m

QPmd

From the equation, we know m = (4, -1, 3)

46.3

26

312

26

)10()14()4(

)3()1()4(

)10,14,4(

)3,1,4(

)5,3,2()3,1,4(

222

222

Pg. 540 # (1, 2, 3, 5)a, 6b, 9

MCV 4U Independent Study 4.6 Distance from a Point to a Plane

Distance from a Point to a Plane

If there is a point ),,( 111 zyxQ off the plane and a point ),,( oooo zyxP on the plane 0 DCzByAx ,

then the distance d from Q to the plane is the projection of QPo onto the normal ),,( CBA .

nontoQPprojd o

)(

n

nQPo

222

111 ),,(),,(

CBA

CBAzzyyxx ooo

222

111 )()()(

CBA

zzCyyBxxA ooo

222

111 )(

CBA

CzByAxCzByAx ooo

Since oP is a point on the plane, it satisfies the plane, so 0 DCzByAx ooo or ooo CzByAxD .

Substituting this into the above equation gives 222

111

CBA

DCzByAxd

The distance from a point (x1, y1, z1 ) to the plane Ax By Cz D 0

is given by the formula

222

111

CBA

DCzByAxd

OR

n

nQPd

o

, where n

is the normal vector of the plane

and Po is a point on the plane and Q is the point whose coordinates are known.

Ex. 1 Find the distance from the point Q (10, 3, -8)

to the plane 01624 zyx .

Point on the plane, Po = (4, 0, 0)

22)1,24()8,3,6(

)8,3,6(

nQP

QP

o

o

OR

80.421

22

)1()2()4(

22

222

n

nQPd

o

80.4

124

16)8()3(2)10(4

222

222

111

CBA

DCzByAxd

Distance between Skew Lines

Skew lines are lines in 3 space that are not parallel and do not intersect. Even though they are not parallel,

they do not intersect because they lie in different planes. They pass each other just like vapour trails left by

two aircraft flying at different altitudes.

Ex. 2 Determine the distance between L1: )3,0,1()5,4,2(1 sr and L2: )0,1,2()1,2,2(2 tr .

To find the distance between skew lines ( lines which do not intersect and are not parallel), we need

a point on both lines ( P1 and P2), the vector P1 P2, and the normal, n, to both lines.

The distance is equal to the scalar projection of 21PP onto n.

Recall: Projection of P1 P2 onto n n

nPP

21

)5,4,2(1 P 21 ddn

)1,2,2(2 P

)4,2,4(21 PP

59.0

46

4

1369

41212

16)3(

)1,6,3()4,2,4(Proj

22221

PPn

OR

)1,2,2(

01363

13

0)5()4(6)2(3

)5,4,2(

063

)1,6,3(

2

1

1

Lonpoint

zyxisL

D

D

Lonpoint

Dzyx

n

59.0

46

4

)1()6()3(

13)1(1)2(6)2(3

222

222

111

CBA

DCzByAxd

Pg. 550 # 1, 2ace, 3, 5