1. “Teach A Level Maths” Vol. 2: A2 Core Modules
53: The Scalar Product of Two Vectors
© Christine Crisp

2. Module C4
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3. Vectors have direction as well as magnitude so it seems strange that we can multiply them together.
There are in fact 2 ways of multiplying vectors.
In this presentation we are going to look at one of these methods: the scalar product.

4. b a b a Suppose the angle between two vectors a and b is .
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection so, this angle . . .
is NOT . a b
We need to reposition b

5. b a b b a Suppose the angle between two vectors a and b is .
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection so, this angle . . .
is NOT . a b
We need to reposition b
then we see that is an obtuse angle. b

6. The result of the scalar product is a scalar quantity not a vector !
Suppose the angle between two vectors a and b is . a b
The scalar product is written as a . b and is defined as
The dot must NEVER be missed out.
The result of the scalar product is a scalar quantity not a vector !
The scalar product is sometimes called the “dot” product.

7. Notice that the scalar product uses the magnitudes, a and b, of the vectors as well as the angle between them, so, we get a different answer for:
different size vectors at the same angle, 8 10 6 10

8. 6 6 10 10 We also have different answers for
the same size vectors at different angles. 6 10 6 10
The scalar product was defined so that the answer is unique to any 2 vectors. It will enable us easily to find the angle between the vectors, even in 3 dimensions.

9. Perpendicular VectorsIf
either a = 0 or
b = 0 or
a = 0 or b = 0 are trivial cases as they mean the vector doesn’t exist. So, we must have
The vectors are perpendicular.

10. SUMMARY
The scalar product is written as a . b and is defined as
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection.
If and
then and the vectors are perpendicular.

11. Exercise
1. Find the scalar product of the following pairs of vectors.
(a)
(b)
(c) b b 4 4 3 b 5 6 4 a a a
2. Find the value of a . a for any non-zero vector a.
3. Find the value of (a) i . i and (b) i . k

12. 1.
(a) 6 4
(b) 5 3 a b
(c)
Solutions:
(a)
(b)
(c)
( not )

13. 2. Find the value of a . a for any non-zero vector a.
Solution: The angle between a vector and itself is
3. i and k are the unit vectors ( magnitude 1 ) along the x- and z- axes respectively, so
(a) i . i = 1 ( since the magnitude of i is one )
(b) i . k = 0 ( since )

14. 2 3 4 -1 -3 2 4 The Scalar Product of 2 Column Vectors Suppose and
Then to form , there are quantities to multiply
However, six of these are perpendicular.
e.g. is along the x-axis and is along the y-axis. 2 4

15. 2 3 -1 4 -3 -1 2 4 The Scalar Product of 2 Column Vectors Suppose and
Then to form , there are quantities to multiply
However, six of these are perpendicular.
e.g. is along the x-axis and is along the y-axis. 2 4
The scalar product of these components is zero.

16. 2 3 -1 4 -3 -1 2 4 The Scalar Product of 2 Column Vectors Suppose and
Then to form , there are quantities to multiply.
However, six of these are perpendicular.
e.g. is along the x-axis and is along the y-axis. 2 4
The scalar product of these components is zero.
The other three multiplications e.g 2 and

17. 2 3 -1 4 -3 -1 2 4 The Scalar Product of 2 Column Vectors Suppose and
Then to form , there are quantities to multiply.
However, six of these are perpendicular.
e.g. is along the x-axis and is along the y-axis. 2 4
The scalar product of these components is zero.
The other three multiplications e.g 2 and
involve parallel components so the angle between them is zero.

18. 2 3 -1 4 -3 -1 The Scalar Product of 2 Column Vectors Suppose and and
Since ,

19. 2 3 -1 4 -3 -1 The Scalar Product of 2 Column Vectors Suppose and and
Since ,

20. 2 3 -1 4 -3 -1 The Scalar Product of 2 Column Vectors Suppose and and
Since ,

21. 2 3 -1 4 -3 -1 The Scalar Product of 2 Column Vectors Suppose and and
Since ,

22. 2 3 -1 4 -3 -1 The Scalar Product of 2 Column Vectors Suppose and and
Since ,

23. SUMMARY
For the scalar product of 2 column vectors,
e.g.
and
we multiply the “tops”,
“middles”
and “bottoms”
and add the results. So,

24. e.g.1 Find the scalar product of the vectors
and
Solution:

25. e.g.2 Show that the vectors
and
are perpendicular.
Solution:

26. Exercise
1. Find the scalar product of the following pairs of vectors.
(a)
and
(b)
and
What can you say about the vectors in part (b) ?

27. (a)
Solutions:
(b)
The vectors are perpendicular.

29. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

30. The result of the scalar product is a scalar quantity not a vector !b
Suppose the angle between two vectors a and b is .
The scalar product is written as a . b and is defined as
The dot must NEVER be missed out.
The scalar product is sometimes called the “dot” product.
The result of the scalar product is a scalar quantity not a vector !

31. Perpendicular VectorsIf
either a = 0 or
a = 0 or b = 0 are trivial cases as they mean the vector doesn’t exist. So, we must have
The vectors are perpendicular.
b = 0 or

32. SUMMARY
The scalar product is written as a . b and is defined as
is defined as the angle which is between the vectors when both point towards, or both away from, the point of intersection.
If and
then and the vectors are perpendicular.

33. SUMMARY
For the scalar product of 2 column vectors,
e.g.
and
we multiply the “tops”,
and add the results. So,
“middles”
and “bottoms”

34. e.g.1 Find the scalar product of the vectors
and
Solution:

35. Solution:
e.g.2 Show that the vectors
and
are perpendicular.