Vectors Lesson 1

1. Each lesson may take a number of classes. VectorsLesson 1 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons

2. Definition of a Vector: A vector has both length and direction. The movement from a to b.

3. Equal Vectors Same length same direction.

4. In the parallelogram abcd The vectors are equal even though they are in different locations.

5. Modulus of a vector: is the length of the vector. Modulus also called absolute value.

6. Negative Vector: is a vector parallel and equal in length, but going in the opposite direction. To get rid of the minus just switch the letters

7. Multiplication of a vector by a scalar.(Number) The 2 ,-1.5 etc are called scalars Negative scalars the arrow goes in the opposite direction

8. Adding Vectors Triangular Law: Put the start of the second vector on the end of the first vector. The result is a vector that goes from the beginning of the first vector to the end of the second vector

9. Adding Vectors (Continued) The Parallelogram Law: Both vectors must have the same starting point. Complete the parallelogram. The diagonal from the starting point is the sum of the two vectors. The Triangular Method and the Parallelogram Method produce the same vector.

10. Example 1 In parts (i) to (iv) the Law is used. In part (v) the Law is used.

11. VectorsLesson 2 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 Mr M Timmons

12. Single Letter Vectors Important Rule The second vector minus the first vector This is a very important result and is used frequently when dealing with vector problems

13. Example 2 Express in terms of and Notice the Rule

14. The Perpendicular vectors The vector The diagram above shows two perpendicular axes, just like the x and y axes.

15. Operations with the Vectors • Adding and Subtracting: you add (or subtract) the parts • and parts separately. You can be asked to show the result on a diagram

16. (ii) Multiplication by a Scalar (iii) The Vector This is the very important result that I talked about earlier & is used frequently when dealing with vector problems

17. Equality of Vectors Ex 2 Ex 3 Here we are solving two simultaneous equations.

18. VectorsLesson 3 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons

19. The Modulus of a vector in terms of Ex 1 The Modulus is the distance from the origin to the point (a,b)

20. The related Perpendicular vector Ex 2 To get the related perpendicular vector we swop the and coefficients and change the sign of the

21. VectorsLesson 4 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons

22. Dot Product (There are two definitions) Angle Between 2 vectors This last formula gives us the angle between any two vectors

23. Perpendicular Vectors By Definition (ii) If the dot product of two vectors is zero, this implies the two vectors are perpendicular to each other. Don’t confuse with

24. Ex 1 Evaluate the following: When the dot product is zero the vectors are perpendicular

25. Ex 2 Find: Here we use the inverse cos button on your calculator to get the angle. Shift cos( 4÷5)=

26. End of Topic Vectors is your option for Paper 2 of the Leaving Certificate. You do need to know this Material.