Vectors are quantities which also have a direction, this is what makes them more than just a scalar
For example: two objects with velocities of 7 m/s and ‑7 m/s are travelling at the same speed but in oppositedirections
A vector quantity is described by both its magnitude and direction
Avectorhascomponents in the direction of the x- , y-, and z- axes
Vector quantities can have positive or negative components
Some examples of vector quantities you may come across are displacement, velocity, acceleration, force/weight, momentum
Displacement is the position of an object from a starting point
Velocity is a speed in a given direction (displacement over time)
Acceleration is the change in velocity over time
Vectors may be given in either 2- or 3- dimensions
Exam Tip
Make sure you fully understand the definitions of all the words in this section so that you can be clear about what your exam question is asking of you
Worked Example
State whether each of the following is a scalar or a vector quantity.
a)A speed boat travels at 3 m/s on a bearing of 052°
b)A garden is 1.7 m wide
c)A car accelerates forwards at 5.4 ms-2
d)A film lasts 2 hours 17 minutes
e)An athlete runs at an average speed of 10.44 ms-1
f)A ball rolls forwards 60 cm before stopping
Vector Notation
How are vectors represented?
Vectors are usually represented using an arrow in the direction of movement
The length of the arrow represents its magnitude
They are written as lowercase letters either in bold or underlined
For example a vector from the point O to A will be written a or a
The vector from the point A to O will be written -a or -a
If the start and end point of the vector is known, it is written using these points as capital letters with an arrow showing the direction of movement
Two vectors are equal only if their corresponding components are equal
Numerically, vectors are either represented using column vectors or base vectors
Unless otherwise indicated, you may carry out all working and write your answers in either of these two types of vector notation
What are column vectors?
Columnvectors are where one number is written above the other enclosed in brackets
In 2-dimensions the top number represents movement in the horizontal direction (right/left) and the bottom number represents movement in the vertical direction (up/down)
A positive value represents movement in the positive direction (right/up) and a negative value represents movement in the negative direction (left/down)
What are base vectors?
Base vectors use i, j and k notation wherei, j and k are unitvectors in the positive x, y, and z directions respectively
This is sometimes also called unit vector notation
A unit vector has a magnitude of 1
In 2-dimensions i represents movement in the horizontal direction (right/left) and j represents the movement in the vertical direction (up/down)
For example:The vector (-4i + 3j) would mean 4 units in the negativehorizontal (x) direction (i.e., left) and 3 units in the positivevertical (y) direction (i.e., up)
In 3-dimensions i represents movement in the x direction (length), j represents movement in the y direction (width) and k represents the movement in the z direction (depth)
For example:The vector (-4i + 3j - k) would mean 4 units in the negativex direction, 3 units in the positivey direction and 1 unit in the negative z direction
As they are vectors, i,j and k are displayed in bold in textbooks and online but in handwriting they would be underlined (i, j and k)
Exam Tip
Practice working with all types of vector notation so that you are prepared for whatever comes up in the exam
Your working and answer in the exam can be in any form unless told otherwise
It is generally best to write your final answer in the same form as given in the question, however you will not lose marks for not doing this unless it is specified in the question
Vectors appear in bold (non-italic) font in textbooks and on exam papers, etc (i.e. F, α ) but in handwriting should be underlined (i.e.F , α)
Worked Example
Parallel Vectors
How do you know if two vectors are parallel?
Two vectors are parallel if one is a scalar multiple of the other
This means that all components of the vector have been multiplied by a common constant (scalar)
Multiplying every component in a vector by a scalar will change the magnitude of the vector but not the direction
Exam Tip
It is easiest to spot that two vectors are parallel when they are in column vector notation
in your exam by writing vectors in column vector form and looking for a scalar multiple you will be able to quickly determine whether they are parallel or not