Chapter 12Vectors
A. Vectors and scalarsB. Geometric operations with vectorsC. Vectors in the planeD. The magnitude of a vectorE. Operations with plane vectorsF. The vector between two pointsG. Vectors in spaceH. Operations with vectors in spaceI. ParallelismJ. The scalar product of two vectors
Opening Problem
An airplane in calm conditions is flying at 800 km/hr due east. A cold wind suddenly blows from the south-west at 35 km/hr, pushing the airplane slightly off course.
Things to think about:a. How can we illustrate the plane’s movement and the wind using a scale diagram?b. What operation do we need to perform to find the effect of the wind on the airplane?c. Can you use a scale diagram to determine the resulting speed and direction of the airplane?
Vectors and Scalars
Quantities which have only magnitude are called scalars.
Quantities which have both magnitude and direction are called vectors.
The speed of the plane is a scalar. It describes its size or strength.
The velocity of the plane is a vector. It includes both its speed and also its direction.
Other examples of vector quantities are:
acceleration force
Displacement momentum
Directed Line Segment Representation
We can represent a vector quantity using a directed line segment or arrow.
The length of the arrow represents the size or magnitude of the quantity, and the arrowhead shows its direction.
For example, farmer Giles needs to remove a fence post. He starts by pushing on the post sideways to loosen the ground.Giles has a choice of how hard to push the post and in which direction. The force he applies is therefore a vector.
If farmer Giles pushes the post with a force of 50 Newtons (N) to the north-east, we can draw a scale diagram of the force relative to the north line.
45o
N50N
Scale: 1 cm represents 25 N
Vector Notation
Geometric vector equality
Two vectors are equal if they have the same magnitude and direction.
Equal vectors are parallel and in the same direction, and are equal in length. The arrows that represent them are translations of one another.
Geometric negative vectors
AB and BA have the same length, but theyhave opposite directions.
A A
B B
Geometric operations with vectors
A typical problem could be:A runner runs east for 4 km and then south for 2 km. How far is she from her starting point and in what
direction?4 km
2 kmx km
N
S
W E
Geometric vector addition
Suppose we have three towns P, Q, and R.A trip from P to Q followed by a trip from Q to R has the same origin and destination as a trip from P to R.
This can be expressed in vector form as the sum PQ + QR = PR.
P
R
Q
“head-to-tail” method of vector addition
To construct a + b:
Step 1: Draw a.
Step 2: At the arrowhead end of a, draw b.
Step 3: Join the beginning of a to the arrowhead end of b. This is vector a + b.
Given a and b asshown, constructa + b.
ab
ab
a
b
a + b
THE ZERO VECTOR
The zero vector 0 is a vector of length 0.
For any vector a: a + 0 = 0 + a = aa +(-a) =(-a) + a = 0.
Find a single vector which is equal to:a. BC + CAb. BA + AE + ECc. AB + BC + CAd. AB + BC + CD + DE
A
E D
C
B
A
E D
C
B
BC + CA = BA
A
E D
C
B
BA + AE + EC = BC
AB + BC + CA = AA = 0
AB + BC + CD + DE = AE
Geometric vector subtraction
To subtract one vector from another, we simply add its negative.
a – b = a +(-b)
a
b
a
b
-b
a - b
For r, s, and t shown, find geometrically:
a. r – s
b. s – t – r
r
s t
r – s
r s
-sr – s
s – t – r
r
s
t
-ts – t – r
For points A, B, C, and D, simplify the following vector expressions:
a. AB – CB
b. AC – BC – DB
a. Since –CB = BC, then AB – CB = AB + BC = AC.
b. Same argument as part a.
AC – BC – DB = AC + CB + BD = AD
Vector Equation
Whenever we have vectors which form a closed polygon, we can write a vector equation which relates the variables.
Find, in terms of r, s, and t:
a. RS
b. SR
c. STO T
SR
r
t
s
a. Start at R, go to O by –r then go to S by s. Therefore RS = -r + s = s – r.
b. Start at S, go to O by –s then go to R by r. Therefore SR = -s + r = r – s.
c. Start at S, go to O by –s then go to T by t. Therefore ST = -s + t = t – s.
Geometric scalar multiplication
If a is a vector and k is a scalar, then ka is also a vector and we are performing scalar multiplication.
If k > 0, ka and a have the same direction.
If k < 0, ka and a have opposite directions.
If k = 0, ka = 0, the zero vector.
Given vectors r and s, construct geometrically:
a. 2r + s
b. r – 3s
r s
a. 2r + s
r s2r
s
2r + s
b. r – 3s
r
s
r
-3s
r – 3s
Vectors in the plane
In transformation geometry, translating a point a units in the x-direction and b units in the y-direction can be achieved using the translation vector
b
a
Base unit vector
All vectors in the plane can be described in terms of the base unit vectors i and j.
a. 7i + 3j
7i
3j
b. -6ic. 2i – 5jd. 6je. -6i + 3jf. -5i – 5j
The magnitude of a vector
v1
v2v
||.
||.
525
3
q
p
jiqp
b
a
findandIf
units
a
34
)5(3||
5
3.
22
p
p
units
q
b
29
)5(2||
5
252.
22
ji
Unit vectors
A unit vector is any vector which has a length of one unit.
are the base unit vectors in
the positive x and y-directions respectively.
1
0
0
1ji and
Find k given that is a unit vector.
k3
1
Knowing that is a unit vector,
then
k3
1
3
22
9
8
19
1
19
1
13
1
2
2
2
22
k
k
k
k
k
Operations with plane vectors
a
a1
a2
a1+b1
b
b1
b2a2+b2
a+b
.7
4
3
1baba
findandIf
Check your answer graphically.
4
5
73
41ba
Algebraic negative vectors
Algebraic vector subtraction
rqp
pq
rp
.
.
5
2
4
1
2
3
b
a
findandqandIf
6
2
24
31. pqa
1
4
542
213. rqpb
Algebraic scalar multiplication
qp
2qp
q
qp
32
1.
.
3.
3
2,
1
4
c
b
a
findFor
2
19
4
3312
1
2342
1
3
23
1
4
2
13
2
1.
5
8
321
224
3
22
1
42.
9
6
3
233.
qp
qp
q
c
b
a
If p = 3i – 5j and q =-i – 2j, find |p – 2q|.
p – 2q = 3i – 5j – 2(-i – 2j) = 3i – 5j + 2i + 4j = 5i – j
26)1(5|2| 22 qp
The vector between two points
O
A
B
a1 b1
a2
b2
Given points A(-1, 2), B(3, 4), C(4, -5) and O(0, 0), find the position vector of:a. B from O
b. B from A
c. A from C
7
5
52
4)1(.
2
4
24
13.
4
3
04
03.
CAc
ABb
OBa
[AB] is the diameter of a circle with center C(-1, 2). If B is (3, 1), find:
a. BC
b. the coordinates of A.
)3,5(
35
1241
1
4
2
1,
2
1
2
1),,(.
1
4
12
31.
isA
banda
banda
b
asoBCCABut
b
a
b
aCAthenbascoordinatehasAIfb
BCa
Vectors in space
3-D point plotter: demos #72 content disk
Recall the length of the diagonal of a rectangular prism (box),or if it’s easier, call it “3-D Pythagorean Theorem”
Illustrate the points:a. A(0, 2, 0)
b. B(3, 0, 2)
c. C(-1, 2, 3)
a. A(0, 2, 0)
b. B(3, 0, 2)
c. C(-1, 2, 3)
The vector between two points
If P is (-3, 1, 2), Q is (1, -1, 3), and O is (0, 0, 0), find:
a. OP
b. PQ
c. |PQ|
21124||
1
2
4
23
11
31
2
1
3
02
01
03
222
PQ
PQ
OP
Vector Equality
ABCD is a parallelogram. A is (-1, 2, 1), B is (2, 0,-1), and D is (3, 1, 4).
Find the coordinates of C.
A(-1,2,1)
B(2,0,-1)
D(3,1,4)
C(a,b,c)
)2,1,6(
,,
2
2
3
4
1
3
],[][
4
1
3
2
2
3
11
20
12
isCSo
candbaforsolve
c
b
a
ABDC
lengthsamethehavethey
DCtoparallelisABSince
c
b
a
DC
AB
Operations with vectors in space
Properties of vectors
Two useful rules are:
|2|,
2
3
1
aa findIf
Using the one of the properties of vectors, we know that
|ka| = |k| |a|Therefore
|2a| = 2|a|
units1422312||2 222 a
Find the coordinates of C and D:
)1,1,0(
1
1
0
1
3
1
2
3
5
2
2
2
1
3
1
32
52
21
.,
isC
ABOAOC
ABACandACOAOC
AB
ABfindtoneedweCofscoordinatethefindTo
)0,4,1(
0
4
1
1
3
1
3
3
5
2
3
3
isD
ABOAOC
ABADandADOAOD
Parallelism
Parallelism Properties
.2
3
2
13
42
12
2
121
3,21,2
3
21
2
,
rrand
ssthen
kkwithkforsolve
krandkks
s
k
r
kparallelareandSince baba
Unit vectors
If a = 3i - j find:
a. a unit vector in the direction of a
b. a vector of length 4 units in the direction of a
c. vectors of length 4 units which are parallel to a.
a. a unit vector in the direction of a
ji
aa
aa
10
1
10
3
10
110
3
1
3
10
1
||
1
1013||, 22
vectorunit
unitsvectoroflengththe
b. a vector of length 4 units in the direction of a
ji
ji
aa
10
4
10
12
310
14
||
14
c. vectors of length 4 units which are parallel to a.
10
4
10
12
10
4
10
12
10
4
10
12
||
144
||
144
||
1
iii
aa
aa
aa
and
parallelarewhichunitslength
unitslength
parallel
Find a vector b of length 7 in the opposite
direction to the vector
1
1
2
a
1
1
2
6
7
1
1
2
6
17
||
177
1
1
2
6
1
1
1
2
112
1
||
1222
b
b
aa
aa
negativeopposite
lengthofvectorunit
vectorunit
The scalar product of two vectors
There are two different types of product involving two vectors:
1. The scalar product of 2 vectors, which results in a scalar answer and has the notation v●w (read “v dot w”).
2. The vector product of 2 vectors, which results in a vector answer and has the notation vΧw (read “v cross w”).
Scalar product
ALGEBRAIC PROPERTIES OF THE SCALAR PRODUCT
qp
qp
qp
andbetweenangletheb
a
findandIf
.
.
:,
2
0
1
1
3
2