Applied Mathematics I (Math 1014B)
CHAPTER ONE
VECTORS AND VECTOR SPACES
1.1. Scalar and vectors in ℝ𝟐 𝒂𝒏𝒅 ℝ𝟑
Definitions: a) A scalar is a physical quantity that is described by its magnitude only.
For example, temperature, length, and speed are scalars because they are completely
described by a number that tells "how much"-say a temperature of 20°C, a length of 5 cm,
or a speed of 10 m/s.
b) A vector is a physical quantity that is described using both magnitude and its direction.
For instance, velocity, displacement, and force are some examples of vectors.
Vectors in ℝ𝟐 𝒂𝒏𝒅 ℝ𝟑
A vector in the plane ℝ2 can be described as 𝑢 ⃗ = (𝑢1 , 𝑢2 ), where 𝑢1 , 𝑢2 ∈ ℝ.
3
Similarly, a vector in the space ℝ can be described as a triple of numbers 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 )
where 𝑣1 , 𝑣2 , 𝑣3 ∈ ℝ.
EQUAL (OR EQUIVALENT) VECTORS
Definition: Two vectors 𝑢 ⃗ 𝑎𝑛𝑑 𝑣 in ℝ𝟐 ⁄ℝ3 are said to be equal (or equivalent) if they have
the same magnitude and direction, and is denoted by 𝑢 ⃗ = 𝑣.
𝟐
⃗ = (𝑢1 , 𝑢2 ) , 𝑣 = (𝑣1 , 𝑣2 ) 𝑖𝑛 ℝ , 𝑢
That is, if𝑢 ⃗ = 𝑣 𝑖𝑓𝑓 𝑢1 = 𝑣1 𝑎𝑛𝑑 𝑢2 = 𝑣2 .
Definitions:
⃗⃗⃗⃗⃗ defined as an arrow whose initial point is at point A and
1. A located vector is a vector 𝐴𝐵
whose terminal point is at B.
Y
b2 B(b1,b2)
a2
A(a1,a2)
a1 b1 X
2. Position vector is a vector whose initial point is at the origin.
Definition (Parallel / collinear Vectors)
Definition Two non – zero vectors 𝑢
⃗ 𝑎𝑛𝑑 𝑣 of the same dimension are said to be parallel or,
alternatively, collinear if at least one of the vectors is a scalar multiple of the other. If one of
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, Applied Mathematics I (Math 1014B)
the vectors is a positive scalar multiple of the other, then the vectors are said to have the
same direction, and if one of them is a negative scalar multiple of the other, then the vectors
are said to have opposite directions. In other words, the two vectors 𝑢
⃗ 𝑎𝑛𝑑 𝑣 are said to be
⃗ // 𝑣 if there exists a scalar c such that⃗⃗⃗𝑢 = 𝑐 𝑣.
parallel, denoted by 𝑢
REMARK : 1. The vector 0 is parallel to every vector v in the same dimension , since it can be
expressed as the scalar multiple 0 = 0v.
2. The zero vectors has no natural direction, so we will agree that it can be assigned any
direction that is convenient for the problem at hand.
⃗ = (𝑢1 , 𝑢2 ) 𝑎𝑛𝑑 𝑐 ∈ 𝑅, we define 𝑐 𝑢
Definition: For any 𝑢 ⃗ = (𝑐𝑢1 , 𝑐𝑢2 ).
⃗ as 𝑐 𝑢
1.2. Vector addition and Scalar multiplication
⃗ = (𝑢1 , 𝑢2 ) 𝑎𝑛𝑑 𝑣 = (𝑣1 , 𝑣2 ) in ℝ2 , we define their sum to
Definition: For any two vectors 𝑢
be ⃗ + 𝑣 = (𝑢1 + 𝑣1 , 𝑢2 + 𝑣2 ).
𝑢
Geometrically, if we represent the two vectors 𝑢 ⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐵𝐶
⃗ 𝑎𝑛𝑑 𝑣 by 𝐴𝐵 ⃗⃗⃗⃗⃗ respectively, then
⃗ + 𝑣 is represented by⃗⃗⃗⃗⃗⃗
𝑢 𝐴𝐶 , as shown in the diagram below:
C
U+V
V
A
B
U
AB BC AC
a) The Triangular Law
D
C
V U+V = V+U
V
A B
U
b) The Parallelogram Law
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, Applied Mathematics I (Math 1014B)
Addition of vectors on the coordinate plane
Let 𝑣 = (𝑣1 , 𝑣2 ) 𝑎𝑛𝑑 𝑤
⃗⃗ = (𝑤1 , 𝑤2 ) then → + → = (𝑣1 + 𝑤1 , 𝑣2 + 𝑤2 ).
𝑣 𝑤
Vector Subtraction The negative of a vector v, denoted by -v, is the vector that has the
same length as v but is oppositely directed, and the difference of v from w, denoted by w - v,
is taken to be the sum w - v = w+ (-v)
Example: Provided that U =(-1,0,1) and V= (2,-1,5 ) ,find each of the following vectors
a) U + V b) 2U c) V -2U
Solution: a) U+V= (-1,0,1) + (2,-1,5 ) = (1,-1,6)
b) 2U = 2 (-1,0,1) = (-2 , 0 , 2)
c) V -2U = (2,-1,5 )-2(-1,0,1)= (4 ,-1 ,3)
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, Applied Mathematics I (Math 1014B)
Properties of Vector addition & Scalar Multiplication
Let 𝑢 ⃗⃗ be vectors in ℝ2 and 𝑐 & 𝑚 are scalars. Then:
⃗ , 𝑣 and 𝑤
⃗ + 𝑣 ∈ ℝ2
a) 𝑢
b) 𝑢
⃗ +𝑣 =𝑣+𝑢
⃗
⃗ + ⃗0 = ⃗0 + 𝑢
c) 𝑢 ⃗ , 𝑤ℎ𝑒𝑟𝑒 0 = (0,0) ∈ ℝ2 .
⃗ =𝑢
d) There exists 𝑤 ∈ ℝ2 such that 𝑢 ⃗⃗ = 0 for every 𝑢 ∈ ℝ2 .
⃗ +𝑤
⃗ + (𝑣 + 𝑤
e) 𝑢 ⃗⃗ ) = (𝑢
⃗ + 𝑣) + 𝑤
⃗⃗
f) 𝑐 (𝑚 𝑢
⃗ ) = (𝑐𝑚)𝑢
⃗
g) (𝑐 + 𝑚)𝑢
⃗ = 𝑐𝑢
⃗ + 𝑚𝑢
⃗
h) 1. 𝑢
⃗ =𝑢
⃗
Remark: The properties described above also hold true for vectors in ℝ3 , where 0 =
(0,0,0) ∈ ℝ3 , replaces the zero vector 0 in ℝ2 .
SUMS OF THREE OR MORE VECTORS
1.3. Dot (Scalar) product, Magnitude of a vector, Angle between two Vectors, Orthogonal
Projection, Direction angles and direction cosines.
1.3.1. Dot (Scalar) Product
Definition: Let 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 ) be a vector in ℝ3 . Then the magnitude (norm) of 𝑣, denoted
by ‖𝑣‖ is defined by:
Page 4
CHAPTER ONE
VECTORS AND VECTOR SPACES
1.1. Scalar and vectors in ℝ𝟐 𝒂𝒏𝒅 ℝ𝟑
Definitions: a) A scalar is a physical quantity that is described by its magnitude only.
For example, temperature, length, and speed are scalars because they are completely
described by a number that tells "how much"-say a temperature of 20°C, a length of 5 cm,
or a speed of 10 m/s.
b) A vector is a physical quantity that is described using both magnitude and its direction.
For instance, velocity, displacement, and force are some examples of vectors.
Vectors in ℝ𝟐 𝒂𝒏𝒅 ℝ𝟑
A vector in the plane ℝ2 can be described as 𝑢 ⃗ = (𝑢1 , 𝑢2 ), where 𝑢1 , 𝑢2 ∈ ℝ.
3
Similarly, a vector in the space ℝ can be described as a triple of numbers 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 )
where 𝑣1 , 𝑣2 , 𝑣3 ∈ ℝ.
EQUAL (OR EQUIVALENT) VECTORS
Definition: Two vectors 𝑢 ⃗ 𝑎𝑛𝑑 𝑣 in ℝ𝟐 ⁄ℝ3 are said to be equal (or equivalent) if they have
the same magnitude and direction, and is denoted by 𝑢 ⃗ = 𝑣.
𝟐
⃗ = (𝑢1 , 𝑢2 ) , 𝑣 = (𝑣1 , 𝑣2 ) 𝑖𝑛 ℝ , 𝑢
That is, if𝑢 ⃗ = 𝑣 𝑖𝑓𝑓 𝑢1 = 𝑣1 𝑎𝑛𝑑 𝑢2 = 𝑣2 .
Definitions:
⃗⃗⃗⃗⃗ defined as an arrow whose initial point is at point A and
1. A located vector is a vector 𝐴𝐵
whose terminal point is at B.
Y
b2 B(b1,b2)
a2
A(a1,a2)
a1 b1 X
2. Position vector is a vector whose initial point is at the origin.
Definition (Parallel / collinear Vectors)
Definition Two non – zero vectors 𝑢
⃗ 𝑎𝑛𝑑 𝑣 of the same dimension are said to be parallel or,
alternatively, collinear if at least one of the vectors is a scalar multiple of the other. If one of
Page 1
, Applied Mathematics I (Math 1014B)
the vectors is a positive scalar multiple of the other, then the vectors are said to have the
same direction, and if one of them is a negative scalar multiple of the other, then the vectors
are said to have opposite directions. In other words, the two vectors 𝑢
⃗ 𝑎𝑛𝑑 𝑣 are said to be
⃗ // 𝑣 if there exists a scalar c such that⃗⃗⃗𝑢 = 𝑐 𝑣.
parallel, denoted by 𝑢
REMARK : 1. The vector 0 is parallel to every vector v in the same dimension , since it can be
expressed as the scalar multiple 0 = 0v.
2. The zero vectors has no natural direction, so we will agree that it can be assigned any
direction that is convenient for the problem at hand.
⃗ = (𝑢1 , 𝑢2 ) 𝑎𝑛𝑑 𝑐 ∈ 𝑅, we define 𝑐 𝑢
Definition: For any 𝑢 ⃗ = (𝑐𝑢1 , 𝑐𝑢2 ).
⃗ as 𝑐 𝑢
1.2. Vector addition and Scalar multiplication
⃗ = (𝑢1 , 𝑢2 ) 𝑎𝑛𝑑 𝑣 = (𝑣1 , 𝑣2 ) in ℝ2 , we define their sum to
Definition: For any two vectors 𝑢
be ⃗ + 𝑣 = (𝑢1 + 𝑣1 , 𝑢2 + 𝑣2 ).
𝑢
Geometrically, if we represent the two vectors 𝑢 ⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐵𝐶
⃗ 𝑎𝑛𝑑 𝑣 by 𝐴𝐵 ⃗⃗⃗⃗⃗ respectively, then
⃗ + 𝑣 is represented by⃗⃗⃗⃗⃗⃗
𝑢 𝐴𝐶 , as shown in the diagram below:
C
U+V
V
A
B
U
AB BC AC
a) The Triangular Law
D
C
V U+V = V+U
V
A B
U
b) The Parallelogram Law
Page 2
, Applied Mathematics I (Math 1014B)
Addition of vectors on the coordinate plane
Let 𝑣 = (𝑣1 , 𝑣2 ) 𝑎𝑛𝑑 𝑤
⃗⃗ = (𝑤1 , 𝑤2 ) then → + → = (𝑣1 + 𝑤1 , 𝑣2 + 𝑤2 ).
𝑣 𝑤
Vector Subtraction The negative of a vector v, denoted by -v, is the vector that has the
same length as v but is oppositely directed, and the difference of v from w, denoted by w - v,
is taken to be the sum w - v = w+ (-v)
Example: Provided that U =(-1,0,1) and V= (2,-1,5 ) ,find each of the following vectors
a) U + V b) 2U c) V -2U
Solution: a) U+V= (-1,0,1) + (2,-1,5 ) = (1,-1,6)
b) 2U = 2 (-1,0,1) = (-2 , 0 , 2)
c) V -2U = (2,-1,5 )-2(-1,0,1)= (4 ,-1 ,3)
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, Applied Mathematics I (Math 1014B)
Properties of Vector addition & Scalar Multiplication
Let 𝑢 ⃗⃗ be vectors in ℝ2 and 𝑐 & 𝑚 are scalars. Then:
⃗ , 𝑣 and 𝑤
⃗ + 𝑣 ∈ ℝ2
a) 𝑢
b) 𝑢
⃗ +𝑣 =𝑣+𝑢
⃗
⃗ + ⃗0 = ⃗0 + 𝑢
c) 𝑢 ⃗ , 𝑤ℎ𝑒𝑟𝑒 0 = (0,0) ∈ ℝ2 .
⃗ =𝑢
d) There exists 𝑤 ∈ ℝ2 such that 𝑢 ⃗⃗ = 0 for every 𝑢 ∈ ℝ2 .
⃗ +𝑤
⃗ + (𝑣 + 𝑤
e) 𝑢 ⃗⃗ ) = (𝑢
⃗ + 𝑣) + 𝑤
⃗⃗
f) 𝑐 (𝑚 𝑢
⃗ ) = (𝑐𝑚)𝑢
⃗
g) (𝑐 + 𝑚)𝑢
⃗ = 𝑐𝑢
⃗ + 𝑚𝑢
⃗
h) 1. 𝑢
⃗ =𝑢
⃗
Remark: The properties described above also hold true for vectors in ℝ3 , where 0 =
(0,0,0) ∈ ℝ3 , replaces the zero vector 0 in ℝ2 .
SUMS OF THREE OR MORE VECTORS
1.3. Dot (Scalar) product, Magnitude of a vector, Angle between two Vectors, Orthogonal
Projection, Direction angles and direction cosines.
1.3.1. Dot (Scalar) Product
Definition: Let 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 ) be a vector in ℝ3 . Then the magnitude (norm) of 𝑣, denoted
by ‖𝑣‖ is defined by:
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