Vectors

1. Vectors

2. A VECTOR? • Describes the motion of an object • A Vector comprises • Direction • Magnitude • We will consider • Column Vectors • General Vectors • Vector Geometry Size

3. NOTE! Label is in BOLD. When handwritten, draw a wavy line under the label i.e. a Column Vectors Vector a 2 up 4 RIGHT COLUMN Vector

4. b Column Vectors Vector b 2 up 3 LEFT COLUMN Vector?

5. n Column Vectors Vector u 2 down 4 LEFT COLUMN Vector?

6. b a d c Describe these vectors

7. Alternative labelling F B D E G C A H

8. k k k k General Vectors A Vector has BOTH a Length & a Direction All 4 Vectors here are EQUAL in Length and Travel in SAME Direction. All called k k can be in any position

9. k General Vectors Line CD is Parallel to AB B CD is TWICE length of AB D A 2k Line EF is Parallel to AB E EF is equal in length to AB C -k EF is opposite direction to AB F

10. k Write these Vectors in terms of k B D 2k F G ½k 1½k E C -2k A H

11. B k D A C Combining Column Vectors

12. C B A Simple combinations

13. Q P R a b O Vector Geometry Consider this parallelogram Opposite sides are Parallel OQ is known as the resultant of a and b

14. Resultant of Two Vectors • Is the same, no matter which route is followed • Use this to find vectors in geometrical figures

15. . Q S S is the Midpoint of PQ. Work out the vector P R a b O Example = a + ½b

16. . Q S S is the Midpoint of PQ. Work out the vector P R a b O Alternatively - ½b = b + a = ½b + a = a + ½b

17. C p M Find BC = + A q B BC BA AC AC= p, AB = q Example M is the Midpoint of BC = -q + p = p - q

18. C p M Find BM = ½BC A q B BM AC= p, AB = q Example M is the Midpoint of BC = ½(p – q)

19. C p M Find AM + ½BC = A q B AM AB AC= p, AB = q Example M is the Midpoint of BC = q + ½(p – q) = q +½p - ½q = ½q +½p = ½(q + p) = ½(p + q)

20. C p M Find AM + ½CB = A q B AC AM AC= p, AB = q Alternatively M is the Midpoint of BC = p + ½(q – p) = p +½q - ½p = ½p +½q = ½(p + q)