You need to be able to recall the definition of a vector as well as how to add, subtract, and multiply with them.
Definitions
A vector is a quantity with both magnitude and direction. For example, "5 metres north" is a vector.
A scalar is a quantity with magnitude only. For example, "5 metres" is a scalar as you don't know in which direction.
Representing vectors
There are different ways to represent vectors:
Column vectors
Vectors are commonly represented in column form. Column vectors are written like this: (ab). This means move a steps right and b steps up.
Bold lowercase letters
Bold lowercase letters are used as a quick way to refer to a vector. For example, you may see a= (11).
Uppercase letters with an arrow
The vector from the point A to the point B can be denoted as AB.
On a graph
On a graph, draw vectors with an arrow to point in the direction it is moving in.
Adding and subtracting vectors
To add and subtract column vectors, add and subtract each row individually.
Example 1
If a=(21) and b=(−63), work out a+b and a−b:
a+b=(21)+(−63)=(2+(−6)1+3)=(−44)
a+b=(−44)
a−b=(21)−(−63)=(2−(−6)1−3)=(8−2)
a−b=(8−2)
Multiplying a vector with a scalar
To multiply a vector with a scalar, multiply each number in the vector by the scalar.
Example 2
If a=(4−1), work out 3a, and −2a:
3a=3(4−1)=(3×43×−1)=(12−3)
3a=(12−3)
−2a=−2(4−1)=(−2×4−2×−1)=(−82)
−2a=(−84)
Vector geometry: arithmetic
Adding, subtracting, and multiplying vectors each have their own geometric interpretations.
Addition
Adding two vectors a and b gives the vector a+b, which represents travelling along the vector a then along the vector b in one journey.
Multiplication by a positive scalar
Multiplying a vector a by a positive scalar (number) k gives the vector ka, which is a vector in the same direction as a, but its length is multiplied by a factor of k.
Negative of a vector/multiplication by −1
The negative of a vector a is denoted as (−a), which means it is the same length as a, but going in the opposite direction.
Subtraction
a−b=a+(−b). So subtracting the vector b from a represents going along the vector a, and then going along the vector −b.
Example 3
In the diagram, find the vector AB:
To get from the point A to the point B, you go from A to O, then from O to B. Mathematically:
AB=AO+OB
From the diagram, OB=b.
If OA=a, then AO=−a because it is going in the opposite direction.