The scalar product (also known as the dot product) is one form in which two vectors can be combined together
The result of taking the scalar product of two vectors is a real number
i.e. a scalar
The scalar product of two vectors gives information about the angle between the two vectors
If the scalar product is positive then the angle between the two vectors is acute (less than 90°)
If the scalar product is negative then the angle between the two vectors is obtuse (between 90° and 180°)
If the scalar product is zero then the angle between the two vectors is 90° (the two vectors are perpendicular)
How is the scalar product calculated?
There are two methods for calculating the scalar product
The scalar product can be used in the second formula to find the angle between the two vectors
What properties of the scalar product do I need to know?
Exam Tip
Whilst the formulae for the scalar product are given in the formula booklet, the properties of the scalar product are not, however they are important and it is likely that you will need to recall them in your exam so be sure to commit them to memory
Worked Example
Angle Between Two Vectors
How do I find the angle between two vectors?
If two vectors with different directions are placed at the same starting position, they will form an angle between them
The two formulae for the scalar product can be used together to find this angle
This is given in the formula booklet
To find the angle between two vectors:
Calculate the scalar product between them
Calculate the magnitude of each vector
Use the formula to find cos θ
Use inverse trig to find θ
Exam Tip
The formula for this is given in the formula booklet so you do not need to remember it but make sure that you can find it quickly and easily in your exam
Worked Example
Perpendicular Vectors
How do I know if two vectors are perpendicular?Worked Example