Vectors I - Mathematics Form 2 Notes

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  • A vector is a quantity with both magnitude and direction, e.g. acceleration velocity and force. A quantity with magnitude only is called scalar quantity e.g. mass temperature and time.

Representation of Vectors

  • A vector can be presented by a directed line as shown below:
    representation of vectors
  • The direction of the vector is shown by the arrow.
  • Magnitude is the length of AB
  • Vector AB can be written as AB or AB
  • Magnitude is denoted by |AB|
  • A is the initial point and B the terminal point

Equivalent Vectors

  • Two or more vectors are said to be equivalent if they have:
    • Equal magnitude
    • The same direction.

Equal vectors - two vectors are equal if they have the same magnitude and direction as shown in the fgure below

equal vectors

Negative vectors - Two vectors are negative of each other if they have the same magnitude and opposite directions as shown in the figure below

negative vectors

Addition of Vectors

  • A movement on a straight line from point A to B can be represented using a vector. This movement is called displacement
  • Consider the displacement from u followed by v
    addition pf vectors
  • The resulting displacement is written as u + v

Zero Vector

zero vector

  • Consider a diplacement from A to B and back to A .The total displacement is zero denoted by O
  • This vector is called a Zero or null vector.
    AB + BA = O
    If a + b = 0 , b = -a or a = - b


Multiplication of a Vector by a Scalar

Positive Scalar

If AB= BC =CD=a


AD = a + a +a =3a

Negative Scalar

  • Subtraction of one vector from another is performed by adding the corresponding negative Vector.
  • That is, if we seek a - b we form a + (-b).
    DA = (- a) + (-a) + (-a)
    = -3a

The Zero Scalar

  • When vector a is multiplied by o, its magnitude is zero times that of a. The result is zero vector.
    a.0 = 0.a = 0

Multiplying a Vector by a Scalar

  • If k is any positive scalar and a is a vector then ka is a vector in the same direction as a but k times longer.
  • If k is negative, ka is a vector in the opposite direction to a and k times longer.

More Illustrations

  • A vector is represented by a directed line segment, which is a segment with an arrow at one end indicating the direction of movement. Unlike a ray, a directed line segment has a specific length.
  • The direction is indicated by an arrow pointing from thetail(the initial point) to the head (the terminal point). If the tail is at point A and the head is at point B, the vector from A to B is written as: 
    notation:| AB|
    (Vectors may also be labeled as a single bold face letter, such as vector v.)
    directed line segment
  • The length (magnitude) of a vector v is written |v|. Length is always a non-negative real number.
  • As you can see in the diagram below, the length of a vector can be found by forming a right triangle and utilizing the Pythagorean Theorem or by using the Distance Formula.
    magnitude of a vector
  • The vector at the right translates 6 units to the right and 4 units upward. The magnitude of the vector is 2√13 from the Pythagorean Theorem, or from the Distance Formula:
    AB| = √[(8−2)2 + (7−3)2] = 2√13
  • The direction of a vector is determined by the angle it makes with a horizontal line.
    direction of a vector
  • In the diagram above, to find the direction of the vector (in degrees) we will utilize trigonometry. The tangent of the angle formed by the vector and the horizontal line (the one drawn parallel to the x-axis) is 4/6 (opposite/adjacent).
    tan ∠ A = 4/6
    tan-1 (4/6) ≈ 33.7o
  • A free vector is an infinite set of parallel directed line segments and can be thought of as a translation. Notice that the vectors in this translation which connect the pre-image vertices to the image vertices are all parallel and are all the same length.
  • You may also hear the terms "displacement" vector or "translation" vector when working with translations.

Position Vector

  • To each free vector (or translation), there corresponds a position vector which is the image of the origin under that translation. Unlike a free vector, a position vector is "tied" or "fixed" to the origin. A position vector describes the spatial position of a point relative to the origin.

Traslation Vector

  • Translation vector moves every point of an object by the same amount in the given vector direction.
  • It can be simply be defined as the addition of a constant vector to every point.
  • Translations and vectorsThe translation at the left shows a vector translating the top triangle 4 units to the right and 9 units downward. The notation for such vector movement may be written as: 
    translation vector

  • Vectors such as those used in translations are what is known as free vectorsAny two vectors of the same length and parallel to each other are considered identical. They need not have the same initial and terminal points.

translation vector 2


The points A (-4 ,4 ) , B (-2 ,3) , C (-4 , 1 ) and D ( - 5 , 3) are vertices of a quadrilateral. If the quadrilateral is given the translation T defined by the vector (5/-3)draw the quadrilateral ABCD and its image under T


Summary on Vectors

Components of a Vector in 2 dimensions:

To get from A to B you would move:

  • 2 units in the x direction (x component)
  • 4 units in the y direction (y component)

The components of the vector are these moves in the form of a column vector.

thus vector AB

Similarly: vector CD

A 2-dimensional column vector is of the form 2 d column vector

summary on vectors 1

summary of vectors 2

Magnitude of a Vector in 2 dimensions:

We write the magnitude of u as |u|

u =  2 d column vector then |u| = √(x2 +y2)

The magnitude of a vector is the length of the directed line segment which represents it.

Use Pythagoras’ Theorem to calculate the length of the vector.

The magnitude of vector u is |u| (the length of PQ)

length of PQ
The length of PQ is written as vector PQ
then magnitude of PQ 1

and so magnitude of PQ 2


  1. Draw a directed line segment representing
  2. and P is (2, 1 ), find co ordinates of Q
  3. P is (1 , 3) and Q is (4, 1 ) find


    vector example 1
    vector solution
    Q is ( 2 + 4, 1 + 3)  Q(6, 4)
    vector solution 2
    vector solution 3


  • A quantity which has magnitude and direction.


  • A quantity which has magnitude only.


Displacement, force, velocity, acceleration.


Temperature, work, width, height, length, time of day.

Past KCSE Questions on the Topic.

  1. Given that 4p − 3q =vectors q1a and p + 2q = vectors q1b  find
      1. p and (3 mks)
      2. |p+2q| (3 mks)
    1. Show that A (1 , -1 ), B (3, 5) and C (5, 1 1 ) are collinear (4 mks)
  2. Given the column vectors a = vectors q2 b = vectors q2b c = vectors q2cand that p = 2a − 1/3b + c
    1. Express p as a column vector (2 marks)
    2. Determine the magnitude of p (1 mark)
  3. Given the points P(−6, −3), Q(−2, −1 ) and R(6, 3) express PQ and QR as column vectors. Hence show that the points P, Q and R are collinear. (3 mks)
  4. The position vectors of points x and y are x = 2i + j − 3k and y=3+ 2j − 2k respectively. Find x y as a column vector (2 mks)
  5. Given that a = vectors q5a b = vectors q5b cvectors q5c and p = 2a + b − 3c . Find |p|(3mks)
  6. The position vectors of A and B are vectors q6a and vectors q6b respectively. Find the coordinates of M which divides AB in the ratio 1 :2. (3 marks)
  7. The diagram shows the graph of vectors EF, FG and GH.
    vectors q7
    Find the column vectors;
    1. EH (1 mk)
    2. |EH| (2mks)
  8. OA = 2i − 4k and OB = −2i + j − k. Find |⃗AB|(2mks)
  9. Find scalars m and n such that
    mvectors q9m + n vectors q9n = vectors q9answer
  10. Given that p = 2i j + k and q = i + j +2k, determine 
    1. p + q(1 mk
    2. 1/2p − 2q│(2 mks)
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