3. Higher Mathematics Unit 3.1 Vectors

4. A B 1.Introduction A vector is a quantity with both magnitude and direction. It can be represented using a direct line segment This vector is named or or u

5. 3 5 2 2.Vectors in 3 - Dimensions 2 5 3

6. -2 4 3

7. 3 2 0

8. 3 -3 -2

9. 3. Finding the components of a Vector from Coordinates

10. P (1, 2) Q (6, 3) 3 2 1 6 6 - 1 5 3 - 2 1

11. S (-2, 1) T (5, 3) 3 1 -2 5 5 - -2 7 3 - 1 2

12. A (-2, -1) B (4, 1) 1 - 1 -2 4 4 - -2 6 1- - 1 2

13. 4.Magnitude

14. 4 4 -3 -3 42 + (-3)2

15. 5.Adding Vectors

16. 7 1 C B 1 2 -6 4 A D Add vectors “ Nose-to-tail”

17. v u + v u Add vectors “ Nose-to-tail”

18. A A -u u B B is the negative of is the negative of

19. v -v u + -v u u - v -2 -4 Add the negative of the vector “ Nose-to-tail”

20. -v v The Zero Vector Back to the start. Gone nowhere

21. 7.Multiplication by a Scalar

22. 2v v 2v has TWICE the MAGNITUDE of v, but v and 2v have the SAME DIRECTION. i.e. They are PARALLEL

23. 8.Position Vectors

24. P (4, 2) p The position vector is denoted by If P has coordinates (x , y , z) then the components of the position vector of P are 4 2 The position vector of a point P is the vector from the origin O, to P.

25. 9.Collinear points

26. NOT collinear E D then the vectors are parallel and have a point in common - namely B - , this makes them collinear B C Collinear A

27. 10.Dividing lines in given ratios“Section Formula”

28. Give up John, they are getting bored!!

29. 11.Unit Vectors

30. A unit vector is any vector whose length (magnitude) is one is a unit vector since The vector

31. There are three special unit vectors:

32. -2 4 3 +3 +4 -2 All vectors can be represented using a sum of these unit vectors

33. 12.Scalar Product

34. The scalar product of the vectors and is defined as: The scalar product (or “dot” product) is a kind of vector “multiplication”. It is quite different from any kind of multiplication we’ve met before. where q is the angle between the vectors, pointing out from the vertex or

35. Calculating the angle between two vectors We have already seen that Rearranging gives And hence we can find the angle between two vectors

36. 3. Perpendicular vectors: • Provided and are non zero then if then so ie and are perpendiculiar 2. If either or then 4. Some important results using the scalar product • The scalar product is a number not a vector