Vectors a a ~ ~ Some terms to know: Vectors: Lines with length (magnitude) and direction O A Vector a ~ a ~

Vectors a ~ | | Zero vector or Null vector No length No direction Magnitude of is represented by a ~ | | Or magnitude of is represented by Direction is represented by column vector Zero vector or Null vector No length No direction

Vectors Q y = +4 P x = +2 Movement parallel to the x-axis to the y-axis P Q y = +4 x = +2

Vectors P Q y = - 4 x = - 2

Equal vectors or equivalent vectors P Q Same length R S Same direction

reverses the direction Vectors Negative vectors P Q Same length R S opposite direction NOTE: Negative sign reverses the direction

Let’s do up Exercise 5A in your foolscap paper Q. 1, 3, 5

Vectors Vector addition Vector subtraction

Triangle law of vector addition Ending point Vectors Triangle law of vector addition R Ending point P Q Starting point Parallelogram law of vector addition P Q R Ending point Starting point S

Let’s do up Exercise 5B in your foolscap paper Q. 3, 4, 6, 7

Vectors Scalar multiplication

Let’s do up Exercise 5C in your foolscap paper Q. 1, 5, 10

Vectors P Q y = - 5 x = 2

Vectors Parallel vectors P Q Same gradient R S

Vectors Gradient of vector P Q y = +4 x = +2

Vectors Collinear vectors Lie on the same straight line P, Q and R are collinear => P, Q and R lie on the same straight line R P P Q

Vectors Eg. Given c = , find (a) |2c| (b) |-2c| (c) 2|c| ~ (a) 2c = 2 |2c| =

Vectors Eg. Given c = , find (a) |2c| (b) |-2c| (c) 2|c| ~ (b) -2c = -2 |-2c| =

Vectors Eg. Given c = , find (a) |2c| (b) |-2c| (c) 2|c| ~ (c) 2|c| =

Therefore, |2c| = |-2c| = 2|c| Vectors Notice: |2c| = 10 units |-2c| = 10 units 2|c| = 10 units Therefore, |2c| = |-2c| = 2|c| In conclusion: |kc| = |-kc| = k|c|, where k is any positive number

Vectors Eg. Given a = and b = , find (a) |a| + |b| (b) |a+b| ~ ~ |a| + |b| a+b |a+b| = Note: |a| + |b| ≠ |a + b|

Vectors Eg. Given a = and b = , find (c) |2b-a| (d) 2|b| + |a| ~ ~ 2b-a 2|b| + |a| |2b-a| = Note: |2b -a| ≠ 2|b| - |a|

Let’s do up Exercise 5D in your foolscap paper Q. 1, 6, 8

Vectors Position vectors: always with respect to origin Can you tell me what is the position vector of P? x y O P

Vectors In general, position vector of any point, say P, is given by y x y O P

Vectors Position vectors y P Q O x

Let’s do up Exercise 5E in your foolscap paper Q. 3, 6, 12