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

2. The Scalar Product of Two Vectors Module C4 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. The Scalar Product of Two Vectors In this presentation we are going to look at one of these methods: the scalar product. 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.

4. The Scalar Product of Two Vectors a b 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... a b is NOT. We need to reposition b

5. The Scalar Product of Two Vectors a b 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... a b is NOT. We need to reposition b b then we see that is an obtuse angle.

6. The Scalar Product of Two Vectors a 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 !

7. The Scalar Product of Two Vectors 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

8. The Scalar Product of Two Vectors the same size vectors at different angles. We also have different answers for 6 10 6 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. The Scalar Product of Two Vectors Perpendicular Vectors If 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

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

12. The Scalar Product of Two Vectors 1. (a) 6 4 (b) 5 3 a b a b (c) a b 4 4 (a) (b) (c) ( not ) Solutions:

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

14. The Scalar Product of Two Vectors The Scalar Product of 2 Column Vectors 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. 24 Suppose and 2 3 4 -3

15. The Scalar Product of Two Vectors 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. 24 The scalar product of these components is zero. and 2 3 4 -3

16. The Scalar Product of Two Vectors The Scalar Product of 2 Column Vectors Suppose and Then to form, there are quantities to multiply. The other three multiplications e.g 2 and 3... However, six of these are perpendicular. e.g. is along the x -axis and is along the y -axis. 24 The scalar product of these components is zero. and 2 3 4 -3

17. The Scalar Product of Two Vectors The Scalar Product of 2 Column Vectors Suppose and Then to form, there are quantities to multiply. The other three multiplications e.g 2 and 3... However, six of these are perpendicular. e.g. is along the x -axis and is along the y -axis. 24 The scalar product of these components is zero. involve parallel components so the angle between them is zero. and 2 3 4 -3

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

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

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

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

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

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

24. The Scalar Product of Two Vectors e.g.1 Find the scalar product of the vectors and Solution:

25. The Scalar Product of Two Vectors Solution: e.g.2 Show that the vectorsand are perpendicular.

26. The Scalar Product of Two Vectors 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. The Scalar Product of Two Vectors (b) The vectors are perpendicular. Solutions: (a)

28. The Scalar Product of Two Vectors

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 Scalar Product of Two Vectors a 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. The Scalar Product of Two Vectors Perpendicular Vectors If 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. The Scalar Product of Two Vectors 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. The Scalar Product of Two Vectors 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. The Scalar Product of Two Vectors e.g.1 Find the scalar product of the vectors and Solution:

35. The Scalar Product of Two Vectors Solution: e.g.2 Show that the vectorsand are perpendicular.