A FIRST COURSE
IN LINEAR ALGEBRA
An Open Text by Ken Kuttler
Rn : Vectors
Lecture Notes by Karen Seyffarth∗
Adapted by
LYRYX SERVICE COURSE SOLUTION
∗
Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
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Rn : Vectors
Page 1/31
What is Rn ?
Notation and Terminology
R denotes the set of real numbers.
R2 denotes the set of all column vectors with two entries.
R3 denotes the set of all column vectors with three entries.
In general, Rn denotes the set of all column vectors with n entries.
Rn : Vectors
What is Rn?
Page 2/31
Scalar quantities versus vector quantities
A scalar quantity has only magnitude; e.g. time, temperature.
A vector quantity has both magnitude and direction; e.g. displacement,
force, wind velocity.
Whereas two scalar quantities are equal if they are represented by the same value,
two vector quantities are equal if and only if they have the same magnitude and
direction.
Rn : Vectors
What is Rn?
Page 3/31
R2 and R3
Vectors in R2 and R3 have convenient geometric representations as position
vectors of points in the 2-dimensional (Cartesian) plane and in 3-dimensional
space, respectively.
Rn : Vectors
What is Rn?
Page 4/31
z
c
R3
R
2
(a, b, c)
y
(a, b)
b
0
0
a
b
x
y
a
The vector
Rn : Vectors
a
.
b
What is Rn?
x


a

b .
The vector
c
Page 5/31
Notation
If P is a point in Rn with coordinates (p1 , p2 , ..., pn ) we denote this by
P = (p1 , p2 , ..., pn ).
If P = (p1 , p2 , . . . , pn ) is a point in Rn , then


p1

−
→ 
 p2 
0P =  . 
 .. 
pn
is often used to denote the position vector of the point.
Instead of using a capital letter to denote the vector (as we generally do
with matrices), we emphasize the importance of the geometry and the
direction with an arrow over the name of the vector.
Rn : Vectors
What is Rn?
Page 6/31
Notation and Terminology
−
→
The notation 0P emphasizes that this vector goes from the origin 0 to the
point P. We can also use lower case letters for names of vectors. In this
−
→
case, we write 0P = p~.
Any vector



~x = 

x1
x2
..
.



 in Rn

xn
is associated with the point (x1 , x2 , . . . , xn ).
Often, there is no distinction made between the vector ~x and the point
(x1 , x2 , . . . ,xn ), and we say that both (x1 , x2 , . . . , xn ) ∈ Rn and
x1
 x2 


~x =  .  ∈ Rn .
 .. 
xn
Rn : Vectors
What is Rn?
Page 7/31
Geometric Vectors in R2 and R3
Let A and B be two points in R2 or R3 .
y
•
•
•
•
B
0
x
−→
AB is the geometric vector from A to B.
−→
A is the tail of AB.
−→
B is the tip of AB.
−→
the magnitude of AB is its length, and is
−→
denoted ||AB||.
A
Rn : Vectors
Geometric Vectors
Page 8/31
Equality of geometric vectors
y
B
−→
• AB is the vector from A = (1, 0).
to B = (2, 2).
−→
• CD is the vector from C = (−1, −1)
to D = (0, 1).
D
0
A
x
−→ −→
• AB = CD because the vectors have
the same length and direction.
C
The fact that the points A and B are different from the points C and D is not
important. For geometric vectors, the location of the vector in the plane (or in
3-dimensional space) is not important; the important properties are its length and
direction.
Rn : Vectors
Geometric Vectors
Page 9/31
Coordinatizing Vectors – Part 1
y
P
B
−
→
0P is the position vector for P = (1, 2),
−
→
1
and 0P =
.
2
0
A
x
−→ −
→
−→
Since AB = 0P, it should be the case that AB =
−→
moving AB so that its tail is at the origin.
1
2
. This can be seen by
A geometric vector is coordinatized by putting it in standard position, meaning
with its tail at the origin, and then identifying the vector with its tip.
Rn : Vectors
Geometric Vectors
Page 10/31
Algebra in Rn
Addition in Rn
Since vectors in Rn are n × 1 matrices, addition in Rn is precisely matrix addition
using column matrices, i.e.,
If u~ and ~v are in Rn , then u~ + ~v is obtained by adding together
corresponding entries of the vectors.
The zero vector in Rn is the n × 1 zero matrix, and is denoted ~0.
Example




1
4
Let u~ =  2  and ~v =  5 . Then,
3
6

 
 

1
4
5
u~ + ~v =  2  +  5  =  7 
3
6
9
Rn : Vectors
Algebra in Rn
Page 11/31
Properties of Vector Addition
~ be vectors in Rn . Then the following properties hold.
Let u~, ~v , and w
1
u~ + ~v = ~v + u~
2
~ = u~ + (~v + w
~)
(~
u + ~v ) + w
3
u~ + ~0 = u~
(existence of an additive identity).
4
u~ + (−~
u ) = ~0
(existence of an additive inverse).
Rn : Vectors
(vector addition is commutative).
Algebra in Rn
(vector addition is associative).
Page 12/31
Scalar Multiplication
Since vectors in Rn are n × 1 matrices, scalar multiplication in Rn is precisely
matrix scalar multiplication using column matrices, i.e., If u~ is a vector in Rn and
k ∈ R is a scalar, then k u~ is obtained by multiplying every entry of u~ by k.
Example


1
Let u~ =  2  and k = 4. Then,
3

 

1
4
k u~ = 4  2  =  8 
3
12
Rn : Vectors
Algebra in Rn
Page 13/31
Properties of Scalar Multiplication
Let u~, ~v ∈ Rn be vectors and k, p ∈ R be scalars. Then the following properties
hold.
1
k(~
u + ~v ) = k u~ + k~v (scalar multiplication distributes over vector addition).
2
(k + p)~
u = k u~ + p~
u
3
k(p~
u ) = (kp)~
u
4
1~
u = u~
Rn : Vectors
(addition distributes over scalar multiplication).
(scalar multiplication is associative).
(existence of a multiplicative identity).
Algebra in Rn
Page 14/31
The Geometry of Vector Addition
1
Vector Equality. The vectors have the same length and direction.
2
The zero vector, ~0 has length zero and no direction.
3
Addition. Let u~, ~v be vectors. Then u~ + ~v is the diagonal of the
parallelogram defined by u~ and ~v , and having the same tail as u~ and ~v .
u~ + ~v
u~
~v
Rn : Vectors
The Geometry of Vector Addition
Page 15/31
Tip-to-Tail Method for Vector Addition
For points A, B and C ,
−→ −→ −→
AB + BC = AC .
C
−
→
AB
−→
AC
−→
BC
−
→
BC
B
A
Rn : Vectors
The Geometry of Vector Addition
−→
AB
Page 16/31
Example
The diagonals of any parallelogram bisect each other.
To see this, denote the parallelogram by its vertices, ABCD.
B
C
M
A
D
• Let M denote the midpoint
−→
of AC .
−−→ −−→
Then AM = MC .
• It now suffices to show
−−→ −−→
that BM = MD.
−−→ −→ −−→ −→ −−→ −−→ −→ −−→
BM = BA + AM = CD + MC = MC + CD = MD.
−−→ −−→
Since BM = MD, these vectors have the same magnitude and direction, implying
−→
that M is the midpoint of BD.
Therefore, the diagonals of ABCD bisect each other.
Rn : Vectors
The Geometry of Vector Addition
Page 17/31
The Geometry of Vector Subtraction
Let u~ and ~v be vectors in R2 or R3 . The vector u~ − ~v = u~ + (−~v ) is obtained from
the parallelogram defined by u~ and ~v by taking the vector from the tip of ~v to the
tip of u~, i.e., the diagonal of the parallelogram, directed towards the tip of u~.
u~
u~ − ~v
~v
−~v
u~
Rn : Vectors
The Geometry of Vector Addition
Page 18/31
Coordinatizing Vectors – Part 2
Let A = (x1 , y1 , z1 ) and B = (x2 , y2 , z2 ) be two points in R3 .
B
z
0
y
A
x
−
→ −→ −
→
We see from the figure that 0A + AB = 0B,

 
x
−→ −
→ −
→  2  
AB = 0B − 0A = y2 −
z2
Rn : Vectors
The Geometry of Vector Addition
and hence
 

x1
x2 − x1
y1  =  y2 − y1  .
z1
z2 − z1
Page 19/31
Length of a Vector
The Distance Between Points
For A = (x1 , y1 , z1 ) and B = (x2 , y2 , z2 ) in R3 , the distance between them is
written d(A, B) and is given by
p
d(A, B) = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
This is called the distance formula
If P = (x2 − x1 , y2 − y1 , z2 − z1), then the distance between the origin and P is
equal to the the distance between points A and B i.e.,
p
d(0, P) = d(A, B) = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
Rn : Vectors
Length of a Vector
Page 20/31
Properties of Distance
Let P and Q be two points in Rn , and d(P, Q) the distance between them. Then
the following properties hold.
1
The distance between P and Q is equal to the distance between Q and P,
i.e., d(P, Q) = d(Q, P).
2
d(P, Q) ≥ 0 with equality if and only if P = Q.
Example
For P = (1,p
−1, 3) and Q = (3, 1,√0), the distance between P and Q is
d(P, Q) = 22 + 22 + (−3)2 = 17.
Rn : Vectors
Length of a Vector
Page 21/31
Length of a Vector
More generally, if P = (p1 , p2 , . . . , pn ) and Q = (q1 , q2 , . . . , qn ) are points in Rn ,
−→
−→
then the distance between P and Q is the length of the vector PQ, written kPQk.
p
−→
d(P, Q) = kPQk = (q1 − p1 )2 + (q2 − p2 )2 + · · · + (qn − pn )2 .
x1
y
If ~x =
∈ R2 ,
x2
X = (x1 , x2 )
~
x
0
x
then the length of the vector ~x is the distance from the origin 0 to the point
X = (x1 , x2 ) given by d(0, X ).
The length of ~x , denoted k~x k, is given by:
q
d(0, X ) = k~x k = x12 + x22 .
Rn : Vectors
Length of a Vector
Page 22/31
The formula for calculating the length of a vector generalizes to Rn : if


x1
 x2 


~x =  .  ∈ Rn ,
 .. 
xn
then
k~x k =
q
x12 + x22 + · · · + xn2 ,
which represents the distance from the origin to the point (x1 , x2 , . . . , xn ).
Rn : Vectors
Length of a Vector
Page 23/31
Example



3
−6
−3
~ =  −1 . Then −2~
Let p~ =
and q
q = (−2)~
q =  2 .
4
−2
4
The lengths of these vectors are
p
√
k~
p k = (−3)2 + 42 = 9 + 16 = 5,

k~
qk =
p
(3)2 + (−1)2 + (−2)2 =
√
9+1+4=
and
k − 2~
qk
=
=
=
=
Rn : Vectors
Length of a Vector
p
(−6)2 + 22 + 42
√
36 + 4 + 16
√
√
56 = 4 × 14
√
2 14 = 2k~
q k.
Page 24/31
√
14,
Unit Vectors
Definition
A unit vector is a vector of length one.
Example
 
    √2 
1
0
0
2 
 0 ,  1 ,  0 , 
 √0 , are examples of unit vectors.
2
0
0
1

2
Example
If ~v 6= ~0, then
1
~v
k~v k
is a unit vector in the same direction as ~v .
Rn : Vectors
Length of a Vector
Page 25/31
Example


−1
√
~v =  3  is not a unit vector, since k~v k= 14. However,
−1
√
2
14
1
 √3 
u~ = √ ~v =  14 
14
2
√
14
is a unit vector in the same direction as ~v , i.e.,
1
1 √
k~
u k = √ k~v k = √
14 = 1.
14
14
Example
~ are nonzero that have
If ~v and w
the same direction, then ~v =
k~
vk
~
k~
wk w ;
k~
vk
~
opposite directions, then ~v = − k~
wk w .
Rn : Vectors
Length of a Vector
Page 26/31
The Geometry of Scalar Multiplication
Scalar Multiplication. If ~v =
6 ~0 and a ∈ R, a 6= 0, then a~v has length
ka~v k = |a| · k~v k, and
I
I
has the same direction as ~v if a > 0;
has direction opposite to ~v if a < 0.
Parallel Vectors. Two nonzero vectors are called parallel if they have the
same direction or opposite directions. It follows that nonzero vectors ~v and
~ are parallel if and only if one is a scalar multiple of the other.
w
Rn : Vectors
The Geometry of Scalar Multiplication
Page 27/31
Problem
Let P = (1, −2, 1),Q = (−3, 0, 5), X = (2, −1, 5) and Y = (4, −2, 3) be points in
−→
−→
−→
−−→
R3 . Is PQ parallel to XY ? Is PX parallel to QY ?
Solution




2
−4
−→
−→
−
→
−→ 
2 , XY =  −1 , and these vectors are parallel if PQ = k XY for
PQ =
−2
4
some scalar k, i.e.,

 





−4
2k
−4
2
 2  = k  −1  or  2  =  −k  .
4
−2k
4
−2
This gives a system of three equations in one variable, which is consistent, and
−→
−→
has unique solution k = −2. Therefore, PQ is parallel to XY .
 


1
7
−→   −−→ 
−→
−−→
PX = 1 , QY = −2 , and these vectors are parallel if PX = `QY for
4
−2
−→
−−→
some scalar `. You will find that no such ` exists, so PX is not parallel to QY .
Rn : Vectors
The Geometry of Scalar Multiplication
Page 28/31
Vector problems and examples
Problem
Find the point, M, that is midway between P1 = (−1, −4, 3) and P2 = (5, 0, −3).
Solution
P1
M
P2
0
−→ −−→ −−→ −−→ 1 −−−→
0M = 0P1 + P1 M = 0P1 + P1 P2
2



−1
6
1
 −4  +  4 
2
3
−6

 



−1
3
2
 −4  +  2  =  −2  .
3
−3
0

=
=
Therefore M = (2, −2, 0).
Rn : Vectors
Vector problems and examples
Page 29/31
Problem
Find the two points trisecting the segment between P = (2, 3, 5) and
Q = (8, −6, 2).
Solution
P
A
B
Q
−
→ −
→
−→
• 0A = 0P + 13 PQ
−
→ −
→
−→
• 0B = 0P + 23 PQ
0


6
−→ 
Since PQ = −9 ,
−3

 
 


 
 

2
2
4
2
4
6
−
→
−
→
0A =  3  +  −3  =  0  and 0B =  3  +  −6  =  −3  .
5
−1
4
5
−2
3
Therefore, the two points are A = (4, 0, 4) and B = (6, −3, 3).
Rn : Vectors
Vector problems and examples
Page 30/31
Example
If ABCD is an arbitrary quadrilateral, then the the midpoints of the four sides of
ABCD are the vertices of a parallelogram.
B
M1
M2
A
C
Let M1
M2
M3
M4
−→
denote the midpoint of AB,
−→
the midpoint of BC ,
−→
the midpoint of CD, and
−→
the midpoint of DA.
M4
M3
D
−−−→ −−−→
It suffices to prove that M1 M2 = M4 M3 .
−−−→
M1 M2
−−→ −−→
−−→ −−→
−−−→
= M1 B + BM2
M4 M3 = M4 D + DM3
−
→
−
→
−→
−→
= 12 AB + 12 BC
= 12 AD + 12 DC
−→
→
1−
= 12 AC
=
2 AC
−−−→ −−−→
Since M1 M2 = M4 M3 , the points M1 , M2 , M3 , M4 are the vertices of a
parallelogram.
Rn : Vectors
Vector problems and examples
Page 31/31