1 introduction to vectors length of the arrow represents

1 Introduction to VectorsA vector is a mathematical object that has magnitude and direc-tion.

Graphically, vectors are represented by arrows. The length of thearrow represents the magnitude, and the arrowhead represents the di-rection of the vector.

AB

C

Figure 1.1. The arrows represent vectors A, B and C. The arrowsabove the symbols indicate that the objects are vectors.

Pre-IB Mathematics: ”Course V: Vectors” c⃝ Kari Eloranta 2017

1 Introduction to Vectors (cont.)A vector does not have a specific position in space. If we move a vectorarrow from one position into another retaining its direction, it stillrepresents the same vector.

A

A

A

Figure 1.2. Because a vector does not have a specific location inspace, all arrows represent the same vector A.


1 Introduction to Vectors (cont.)The magnitude of a vector is denoted by modulus signs. For ex-ample, if the magnitude of vector A is three units, we write |A| = 3.

Vectors A and B are equal (A = B), if they are equal in magnitude andpoint at the same direction.

Because vectors are represented by arrows, the magnitude is oftenreferred to as the length of the vector.

The beginning of a vector arrow is called the tail and end of thearrow the tip of the vector.

tailtip


1.1 Zero VectorNext we define a couple of concepts that are important in vectoralgebra. First of all, we need to define a zero vector:

A zero vector 0 is a vector that has zero magnitude and undefineddirection.

A zero vector is just a point in a vector space, such as ℜ2.

A

B

C

Figure 1.3. Zero vectors are represented by points,A = B = C = 0.


1.2 Opposite VectorLet A be a non-zero vector. The opposite vector of A is denotedby −A. It is the vector that is equal in magnitude, but opposite indirection to A.

A

−A

Figure 1.4. A representation of vector A and opposite vector −A.


1.3 Parallel VectorsNon-zero vectors A and B are parallel (A||B), if they point at thesame or opposite direction.

By definition, the zero vector is parallel to all vectors.If the vectors point at the same direction, we write A ↑↑ B. If theypoint at opposite directions, we write A ↑↓ B.

A

BC

Figure 1.5. Examples of parallel vectors. A ↑↑ B and A, B ↑↓ C.


1.3 Parallel Vectors (cont.)In mathematics, in relation to vectors, real numbers are calledscalars.Let A be a non-zero vector and k a scalar (real number) such thatk = 0. Then, the vector kA is a scalar multiple of A.

The magnitude of kA is

|kA| = |k||A|.

If

k > 0, then kA points at the same direction than A (kA ↑↑ A),

k < 0, then kA is opposite to A (kA ↑↓ A).


1.3 Parallel Vectors (cont.)ParallelismLet A and B be non-zero vectors. Vectors A and B are parallel if,and only if, there is a real number k = 0 such that A = kB.That is,

A||B ⇔ A = kB, k = 0 (1)

A ↑↑ B ⇔ A = kB, k > 0

A ↑↓ B ⇔ A = kB, k < 0


Exercise 258 on page 55:Show that vectors a and b point at the same direction, when3(a + b) = 5a− b.


1.4 Non-parallel Vectors

Vectors A and B are non-parallel (A ∦ B), if they are not parallel.

A

B

Figure 1.6. An example of non-parallel vectors, A ∦ B. There isno real number k such that B = kA (B = kA for all k ∈ R).


2 Vector SumTo be able to calculate with vectors, we need to define vector addi-tion and subtraction. First, we learn how to add and subtract vec-tors graphically. Later in the course we learn how the add andsubtract vectors in algebraic form.

Let A and B be vectors. The sum of the vectors is denoted byA + B.

We can determine the sum of the non-zero vectors A and B graph-ically by moving the vector B such that its tail coincides with thetip of the vector A. Then, the sum of the vectors is represented bythe arrow that starts from the tail of A and ends at the tip of B.


A

tail of B

tip of A

B

A

B

A + B

The sum vector A + B goes from tail of A to tip of B

Figure 2.1. The green arrow represents the sum A+ B. All arrowsthat are equal with A + B represent the same vector.


3 Vector SubtractionLet A and B be non-zero vectors. The difference of the vectors isdenoted by A − B. The difference of A and B is defined as theaddition of opposite vector −B

A− B = A + (−B)

We can determine the difference of non-zero vectors A and B graph-ically by constructing the opposite vector −B, and moving it suchthat its tail coincides with the tip of the vector A. Then, the differ-ence of the vectors is represented by the arrow that starts from thetail of A and ends at the tip of −B.


A

B −B

A −B

A− B

Figure 3.1. The green arrow represents the difference A − B =

A + (−B). All arrows that are equal with A − B represent thesame vector.


There is also an alternative way of determining the difference oftwo vectors.

To determine the difference of vectors A and B, you may firstmove the vector arrows so that their tails are at the same point.Then, if you draw an arrow from the tip of B to the tip of A, thearrow represents the vector A− B. If you draw an arrow from thetip of A to the tip of B, the arrow represents the vector B− A.

The method above is very convenient, when the representations ofvectors A and B already share a common starting point, and youhave to express the third vector connecting the tips of the vectorsin terms of A and B, as we see on the following page.


B

A

A− B

B

A

B− A

Figure 3.2. On the left −B+A = A−B, and on the right −A+B =

B− A.


2.5 Base VectorsWhen A and B are non-parallel vectors on a two dimensionalplane, any other vector C on the plane can be expressed in termsof A and B, as illustrated on page 57 in the book.

Let A and B be non-zero vectors such that A = kB, where k = 0is a real number. Then, any other vector C can be expressed as alinear combination of A and B. That is,

C = rA + sB (2)

where r, s ∈ R.


2.5 Base Vectors (cont.)Linear combination means that we multiply the variables by scalarsand sum the results. It is an important mathematical term.Vectors A and B in Equation 2 are said to form a base in a twodimensional vector space (plane). The vectors are called base vectorsof the vector space.

When we express vector C as a linear combination of A and B,

we divide C into parallel components to A and B. Equation 2 is acomponent representation of C.

Let vectors A and B form a base. Then

rA + sB = tA + uB ⇔ r = t and s = u. (3)


2.6 Scalar ProductThe types of multiplication for vectors are the scalar product (dotproduct) and the vector product. We study scalar product in thiscourse, and vector product in the IB course.

The scalar product of non-zero vectors A and B is defined as

A · B = |A||B| cosα (4)

where α is the angle between the vectors A and B.

If A = 0 or B = 0 then the scalar product is defined as A · B = 0.


Note that the result of a scalar product is a scalar (real number),not a vector.

The multiplication sign is ”·” not ”×”, and it cannot be omitted asin real variables where x · y is usually written as xy.

The expression A · B is read as ”a dot b” not ”a times b,” becausethe scalar product is not an ordinary multiplication.

The scalar product is important in several fields of mathematicsand physics. For example in physics, the work done W by a con-stant force F is calculated by W = F · d where d is displacement(the scalar product of force and displacement vectors, Topic 2.3).


2.6 Angle α and Scalar ProductThe angle α between the non-zero vectors A and B can be solvedfrom

cosα =A · B|A||B|

, A, B = 0 (5)

Vectors A and B are perpendicular when

α = 90◦ ⇔ A · B = 0 ⇔ A⊥B (6)

where α is the angle between the vectors A and B.


The following laws apply to scalar product

• A · B = B · A (commutative law)

• A · (B + C) = A · B + A · C (associative law)

• sA · rB = srA · B, s, r ∈ R (multiplication by scalars)

Exercise 2.1. Prove the commutative law above.


2.6 Scalar Product as a Projection

Let A and B be non-zero vectors. Then,|B|∥ := |B| cosα is the scalar projection of B in the direction of A (thesymbol := means ”is defined as”).

B

A|B|cosα


The scalar product can be understood as a measure of parallelismof two vectors.

The scalar product is at maximum when A ↑↑ B (|A||B| cosα =

|A||B| cos 180◦ = |A||B| × 1 = |A||B|).

At minimum when A ↑↓ B (|A||B| cosα = |A||B| cos (−180◦) =

|A||B| × −1 = −|A||B|).

Zero when A⊥B (|A||B| cosα = |A||B| cos 90◦ = |A||B| × 0 = 0).

The scalar product

A · A = |A|2 (7)

because A · A = |A||A| cos 0 = |A|2 · 1 = |A|2.


3 Vectors in xy-coordinate systemWe can represent vectors in a plane by the xy-coordinate system.For that we use perpendicular unit vectors i and j.

i is a unit vector that points at the positive x-direction.j is a unit vector that points at the positive y-direction.

1

2

−1

1 2−1−2x

y

i

j


Because i and j are perpendicular vectors, they form a base in xy-coordinate system.

Any vector A in xy-coordinate system can be expressed in termsof base vectors i and j such that

A = xi + yj (8)

where x and y are real numbers.

Since i and j are unit vectors, |i| = |j| = 1.

You should note that i · j = j · i = 1×1× cos 90◦ = 0, and i · i = j · j =1× 1× cos 0◦ = 1.

i and j form a so-called orthonormal base in xy-plane (from orthogo-nal meaning perpendicular and normal normalized to unit length)


1

2

3

−1

−2

−3

1 2 3−1−2−3−4x

y2i

−2jA

A

A

Figure 3.2. Vectors do not have any specific position in xy−plane.All three arrows represent the same vector A = xi+ yj = 2i+−2j.


Let P = (x, y), where x, y ∈ R, be a point in the xy-plane. Then,the position vector of P is defined as a vector arrow that emanatesfrom origin O and ends at point P .

In terms of base vectors i and j, the position vector is−→OP = xi + yj (9)

where x and y are the x and y coordinates of point P = (x, y).

The length of−→OP is the distance between the origin O and

point P

|−→OP | =

√x2 + y2 (10)


The concept of position vector is fundamental in physics wheredisplacement is defined as the change in position (the first subtopicin Topic 5.1 in IB mathematics).During this course we learn how to construct lines and planes us-ing position vectors.


1

2

3

−1

1 2 3 4−1−2x

y

−→OP

P = (4, 3)

O

xi

yj

Figure 3.1. A position vector representation, where point P =

(4, 3) and−→OP = xi + yj = 4i + 3j. The length of

−→OP is |

−→OP | =√

x2 + y2 =√42 + 32 =

√25 = 5.

−→OP is an example of a displace-

ment vector (in artificial units).


3.6 Vector between Two Points

We can construct a vector using two points in xy-plane. In such acase we talk about the vector between the points, even though avector does not have a specific position in space.

Let a =−→OA and b =

−−→OB be the position vectors of points

A = (x1, y1) and B = (x2, y2). The directed line segment from A toB represents the vector

−→AB = b + (−a) = b− a = (x2 − x1)i + (y2 − y1)j. (11)


The distance between A and B is then

|−→AB| =

√(x2 − x1)2 + (y2 − y1)2 (12)


1

2

3

−1

−2

−3

−4

−5

1 2 3 4−1−2−3−4x

y

a = −→OA

b=−−→OB

O

A = (x1, y1)

B = (x2, y2)

−→AB = b− a

Figure 3.3. In the figure A = (−4, 1) and B = (4, 3). Vector−→AB =

b − a = (x2 − x1)i + (y2 − y1)j = (4 − (−4))i + (3 − 1)j = 8i + 2j.The length of the vector (and the distance between the points) is|−→AB| =

√(x2 − x1)2 + (y2 − y1)2 =

√82 + 22 =

√68 = 2

√17.


3.6 Scalar Product in xy-coordinate system

If vectors A = x1i + y1j and B = x2i + y2j, the scalar product is

A · B = x1x2 + y1y2. (13)

Exercise 3.1. Prove Equation 13.


1 introduction to vectors length of the arrow represents

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