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