The Scalar ('Dot') Product
What is the scalar product?
- The scalar product (also known as the dot product) is one form in which two vectors can be combined together
- The scalar product between two vectors a and b is denoted
- 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 most common method used to find the scalar product between the two vectors v and w is to find the sum of the product of each component in the two vectors
- Where and
- This is given in the formula booklet
- The scalar product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle between them
- Where θ is the angle between v and w
- The two vectors v and w are joined at the start and pointing away from each other
- 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?
- If two vectors, v and w, are parallel then the magnitude of the scalar product is equal to the product of the magnitudes of the vectors
- This is because cos 0° = 1 and cos 180° = -1
- If two vectors are perpendicular the scalar product is zero
- This is because cos 90° = 0
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
Calculate the scalar product between the two vectors and using:
i)
the formula ,
ii)
the formula , given that the angle between the two vectors is 66.6°.
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
Calculate the angle formed by the two vectors and .
Perpendicular Vectors
How do I know if two vectors are perpendicular?
- If the scalar product of two (non-zero) vectors is zero then they are perpendicular
- If then v and w must be perpendicular to each other
- Two vectors are perpendicular if their scalar product is zero
- The value of cos θ = 0 therefore |v||w|cos θ = 0
Worked Example
Find the value of t such that the two vectors and are perpendicular to each other.