Investigative approach to discover the scalar product

1. Inquiry in DP Mathematics Structured Jennifer Wathall Page 1
Scalar Products
Here is an example for a pair of
vectors
5
1
.
3
3
gives a product of 18
Why do you think this is called the scalar product?
Vectors all have two basic properties:
A systematic exploration of the scalar product should look at each of these in turn.
First draw pairs of vectors where the directions stay the same, but the magnitude changes. This will
involve keeping the angle between your pairs constant but changing the length.
For the vectors !
𝒂 𝟏
𝒂 𝟐
! 𝒂𝒏𝒅 !
𝒃 𝟏
𝒃 𝟐
!
The Scalar Product is defined as a1b1 + a2b2
Can you find a relationship between the
magnitude and the scalar product?

2. Inquiry in DP Mathematics Structured Jennifer Wathall Page 2
Now try keeping the magnitude the same, but changing the direction. Try keeping one vector
constant but rotating the second one to see what happens to the scalar product.
Can you sketch a graph of how the product varies as the angle varies from 00 to 3600?
What does the graph look like?
When is it positive, and when negative?
When is it zero?
When is it at its largest and when is it at
its smallest?
What generalisations can you make about the scalar product and the magnitude
and the direction of two vectors?

3. Inquiry in DP Mathematics Structured Jennifer Wathall Page 3
Extension
Find the length c in two
different ways. One way
should use the angle θ,
and one should not. Now
find a formula for the
scalar product of two
unit vectors.