Scalars and Vectors
Scalars - any quantity that has a magnitude, (Size), but no direction.
Example: Age (18 years old), Length (1.8 m), Temperature (22˚ C)
Vectors - any quantity that has a magnitude, and a direction.
Example: Displacement (200 m [up]), Velocity (90 km/h [east]) Note: Directions are put in square [ ] brackets.
Vectors are drawn as arrows with their given direction, and magnitude.
N
100 km [E]
W
50 km [N]
E
S
Vectors can be given various directions... some more formal than others:
Standard Directions:
Compass Directions:
Complex Directions:
[up], [down], [left], [right], [forwards], [backwards]
[N], [E], [S], [W]
[N 20◦ E], [W 35◦ S]
POSITIVE VECTORS: [up] , [right], [forwards], [N], [E]
NEGATIVE VECTORS: [down], [left], [backwards], [S], [W]
Determining Complex Directions:
N
N
◦
100 km [N20 E]
W
W
E
100 km [W35◦ S]
E
S
S
“Aim North & turn 20◦ to the East”
“Aim West & turn 35◦ to the South”
Working with Vectors
Negative Vectors
In order to solve problems, we need to know how to add vectors.
Two things to remember...
1.
Vectors can only be added "head to tail".
This means that the tail, (end without the arrow), of the second
vector must continue from the head, (end with the arrow), of the
vector before it.
2.
Vectors can be moved around, as long as you don't change their
direction or length.
There are only 3 cases that we can run into with addition. They are…
1) Adding Collinear (Parallel) Vectors
2) Adding vectors at right angles to each other
3) Adding vectors that are at odd angles to each (angles other than
90° or 180°)
A negative vector is the almost
exactly like the original, only in the
opposite direction.
Example:

d  20m [E30o N] ,

then  d  20m [W 30o S]
If

+ d

- d
The negative flips the vector.
1) Adding 2 Collinear (Parallel) Vectors
This is pretty easy... we just need to add, (or subtract), the vectors like they're normal numbers.
Two Vectors in the…
Same Direction

a = 5 km
a = 5 km [E]

b = 10 km
b = 10 km [E]


a + b = 15 km [E]


a + b = 15 km
Two Vectors in the…
Opposite Direction * Remember that subtracting a number is the same thing as adding its negative.



c
c = 12 m [E]


d = 3 m [W]
= - 3 m [E]
d


c + d = 12 m [E] + (-3 [E])
= 12 m [E] -3 [E]
=
9 m [E]

c+d
2) Adding 2 Vectors at Right Angles
Find the magnitude of the new vector with the Pythagorean Theorem
Find the direction of the new vector using trig, (SOHCAHTOA).

a = 4 km [E]

a +b

x
b
b = 3 km [N]
3
θ

a
x2 = 32 + 42
= 9 + 16
x2 = 25
x = 5
4


tan θ = 3 / 4
tan θ = 0.75
θ = 37◦
◦
Therefore a + b = 5 km [E 37 N]
Practise Problems for 1) Parallel Vectors
and 2) Right Angle Vectors:
Using the following vectors, find the addition of each set of vectors listed below.

d  11.0 m [W]

e  19.0 m [N]

f  37.0 m [S]

a  25.0 m [E]

b  15.0 m [E]

c  32.0 m [W]


1. a  b


6. c + e
ANSWERS:
1) 40 m [E]


2. c  d

7.

b+e
2) 43 m [W]

3) 7 m [W]

3. a  c

8.

b–e
4) 18 m [S]


4. e  f

9.

c–e
5) 48.9 m [S 41˚ W]


5. c  f

10.

c+a