1. Vectors (1) Units Vectors Units Vectors Magnitude of Vectors Magnitude of Vectors

2. A B The vector AB The vector a a Notation ( ) 8585 … as a column vector 8 across, 5 up

3. The vector a a The vector 2 a a Notation Is twice as long as a, but in the same direction 2a2a a a This bit is a scaler

4. Displacement “a measure of distance and direction” 60 o 50m N An object moves 50m at 60 o to the East-West x-axis How far East has it gone? How far North has it gone? N E Cos 60 o = adj/hyp = E/50 E = 50 cos 60 o = 25m Sin 60 o = opp/hyp = N/50 N = 50 sin 60 o = 43.3m (1 d.p.) This can be expressed as a column vector:- Displacement = [ ] 25 43.3

5. Unit Vectors (1) i i is the unit vector in the x-direction j j is the unit vector in the y-direction i = [ ] 1010 j = [ ] 0101 All vectors can be expressed as a linear combination of these 2 vectors [ ] = 25 + 43.3 e.g. displacement = 25 43.3 [ ] 1010 0101

6. Unit Vectors (2) i = [ ] 1010 j = [ ] 0101 All vectors can be expressed as a linear combination of these 2 vectors [ ] = 25 + 43.3 e.g. displacement = 25 43.3 [ ] 1010 0101 = 25 i + 43.3 j This is the standard way displacement vectors are presented

7. Magnitude of a vector The displacement of a boat is given by :- -10 i + 15 j What is it’s magnitude ? 10 15 -10 i + 15 j By Pythagoras, the magnitude =  (15 2 + 10 2 ) =  325 = 18.0 (1 d.p.) The displacement is 18.0m

8. Magnitude a = -10 i + 15 j a -10 i + 15 j By Pythagoras, the magnitude =  (15 2 + 10 2 ) =  325 = 18.0 (1 d.p.) a is notation for magnitude a = 18.0

9. 4 3 10 a Magnitude of a 3D Vector (1) x y z o

10. x y z 4 3 10 a Magnitude of a 3D Vector (2) o B 3 By Pythagoras, OB =  (3 2 + 4 2 )

11. x y z 4 3 10 a Magnitude of a 3D Vector (3) o B 3 By Pythagoras, OB =  (3 2 + 4 2 ) A  (3 2 + 4 2 ) By Pythagoras, OA 2 = AB 2 + OB 2 AB 2 = 10 2 OB 2 = (3 2 + 4 2 ) OA 2 = 10 2 + 3 2 + 4 2 OA =  (10 2 + 3 2 + 4 2 ) |a| = OA = 11.2

12. x y z r s t a Magnitude of a 3D Vector (General) o |a| =  (r 2 + s 2 + t 2 ) “the magnitude is the square root of the sum of the squares of the 3 components.” [Pythagoras in 3D]