1. Vectors Lesson 1 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons Sunday, 11 October 2015 Each lesson may take a number of classes.

2. The movement from a to b. Definition of a Vector: A vector has both length and direction. Sunday, 11 October 2015

3. Same length same direction. Equal Vectors Sunday, 11 October 2015

4. The vectors are equal even though they are in different locations. In the parallelogram abcd Sunday, 11 October 2015

5. Modulus also called absolute value. Modulus of a vector: is the length of the vector. Sunday, 11 October 2015

6. To get rid of the minus just switch the letters Negative Vector: is a vector parallel and equal in length, but going in the opposite direction. Sunday, 11 October 2015

7. Negative scalars the arrow goes in the opposite direction Multiplication of a vector by a scalar.(Number) The 2,-1.5 etc are called scalars Sunday, 11 October 2015

8. The result is a vector that goes from the beginning of the first vector to the end of the second vector Adding Vectors Triangular Law: Put the start of the second vector on the end of the first vector. Sunday, 11 October 2015

9. The Triangular Method and the Parallelogram Method produce the same vector. 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. Sunday, 11 October 2015

10. In parts (i) to (iv) the Law is used. In part (v) the Law is used. Example 1 Sunday, 11 October 2015

11. Vectors Lesson 2 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 Mr M Timmons Sunday, 11 October 2015

12. This is a very important result and is used frequently when dealing with vector problems Single Letter Vectors Important Rule The second vector minus the first vector Sunday, 11 October 2015

13. Notice the Rule Example 2 Express in terms of and Sunday, 11 October 2015

14. The diagram above shows two perpendicular axes, just like the x and y axes. The Perpendicular vectors The vector Sunday, 11 October 2015

15. You can be asked to show the result on a diagram Operations with the Vectors (i)Adding and Subtracting: you add (or subtract) the parts and parts separately. Sunday, 11 October 2015

16. This is the very important result that I talked about earlier & is used frequently when dealing with vector problems (ii) Multiplication by a Scalar (iii) The Vector Sunday, 11 October 2015

17. Here we are solving two simultaneous equations. Equality of Vectors Ex 3 Sunday, 11 October 2015 Ex 2

18. Vectors Lesson 3 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons Sunday, 11 October 2015

19. The Modulus is the distance from the origin to the point ( a,b ) The Modulus of a vector in terms of Sunday, 11 October 2015 Ex 1

20. To get the related perpendicular vector we swop the and coefficients and change the sign of the The related Perpendicular vector Sunday, 11 October 2015 Ex 2

21. Vectors Lesson 4 Leaving Certificate Ordinary Level Option St. Joseph’s CBS Maths Department Fairview Dublin 3 M Timmons Sunday, 11 October 2015

22. This last formula gives us the angle between any two vectors Dot Product (There are two definitions) Sunday, 11 October 2015 Angle Between 2 vectors

23. Don’t confuse with Perpendicular Vectors Sunday, 11 October 2015 By Definition (ii) If the dot product of two vectors is zero, this implies the two vectors are perpendicular to each other.

24. When the dot product is zero the vectors are perpendicular Ex 1 Sunday, 11 October 2015 Evaluate the following:

25. Here we use the inverse cos button on your calculator to get the angle. Shift cos( 4÷5)= Ex 2 Sunday, 11 October 2015 Find:

26. Vectors is your option for Paper 2 of the Leaving Certificate. You do need to know this Material. End of Topic Sunday, 11 October 2015