The document discusses the key differences between scalar and vector quantities, explaining that scalars only have magnitude while vectors have both magnitude and direction, and provides examples of each. It then focuses on methods for representing and adding vectors graphically and mathematically, including tip-to-tail addition, component methods, and subtraction by adding the opposite vector.

1. Vector & Scalar Quantities

2. Characteristics of a Scalar Quantity
 Only has magnitude
 Requires 2 things:
1. A value
2. Appropriate units
Ex. Mass: 5kg
Temp: 21° C
Speed: 65 mph

3. Characteristics of a Vector Quantity
 Has magnitude & direction
 Requires 3 things:
1. A value
2. Appropriate units
3. A direction!
Ex. Acceleration: 9.8 m/s2 down
Velocity: 25 mph West

4. More about Vectors
 A vector is represented on paper by an arrow
1. the length represents magnitude
2. the arrow faces the direction of motion
3. a vector can be “picked up” and moved on
the paper as long as the length and
direction
its pointing does not change

5. Graphical Representation of a Vector
The goal is to draw a mini version of the vectors to give
you an accurate picture of the magnitude and
direction. To do so, you must:
1. Pick a scale to represent the vectors. Make it simple
yet appropriate.
2. Draw the tip of the vector as an arrow pointing in
the appropriate direction.
3. Use a ruler & protractor to draw arrows for
accuracy. The angle is always measured from the
horizontal or vertical.

6. Understanding Vector Directions
To accurately draw a given vector, start at the second direction
and move the given degrees to the first direction.
N
30° N of E
W E
Start on the East
origin and turn 30° to
the North
S

7. Graphical Representation Practice
 5.0 m/s East
(suggested scale: 1 cm = 1 m/s)
 300 Newtons 60° South of East
(suggested scale: 1 cm = 100 N)
 0.40 m 25° East of North
(suggested scale: 5 cm = 0.1 m)

8. Graphical Addition of Vectors
Tip-To-Tail Method
1. Pick appropriate scale, write it down.
2. Use a ruler & protractor, draw 1st vector to scale in
appropriate direction, label.
3. Start at tip of 1st vector, draw 2nd vector to scale,
label.
4. Connect the vectors starting at the tail end of the 1st
and ending with the tip of the last vector. This
= sum of the original vectors, its called the resultant
vector.

9. Graphical Addition of Vectors (cont.)
Tip-To-Tail Method
5. Measure the magnitude of R.V. with a ruler.
Use your scale and convert this length to its
actual amt. and record with units.
6. Measure the direction of R.V. with a
protractor and add this value along with the
direction after the magnitude.

10. Graphical Addition of Vectors (cont.)
5 Km
Scale: 1 Km = 1 cm
3 Km
Resultant Vector (red) = 6 cm,
therefore its 6 km.

11. Vector Addition Example #1
 Use a graphical representation to solve the
following: A hiker walks 1 km west, then 2
km south, then 3 km west. What is the sum of
his distance traveled using a graphical
representation?

12. Vector Addition Example #1 (cont.)
Answer = ????????

13. Vector Addition Example #2
 Use a graphical representation to solve the
following: Another hiker walks 2 km south
and 4 km west. What is the sum of her
distance traveled using a graphical
representation? How does it compare to hiker
#1?

14. Vector Addition Example #2 (cont.)
Answer = ????????

15. Mathematical Addition of Vectors
 Vectors in the same direction:
Add the 2 magnitudes, keep the direction
the same.
Ex. + =
3m E 1m E 4m E

16. Mathematical Addition of Vectors
 Vectors in opposite directions
Subtract the 2 magnitudes, direction is the
same as the greater vector.
Ex.
4m S + 2m N = 2m S

17. Mathematical Addition of Vectors
 Vectors that meet at 90°
Resultant vector will be hypotenuse of a
right triangle. Use trig functions and
Pythagorean Theorem.

18. Mathematical Subtraction of Vectors
 Subtraction of vectors is actually the addition
of a negative vector.
 The negative of a vector has the same
magnitude, but in the 180° opposite direction.
Ex. 8.0 N due East = 8.0 N due West
3.0 m/s 20° S of E = 3.0 m/s 20° N of W

19. Subtraction of Vectors (cont.)
 Subtraction used when trying to find a change
in a quantity.
 Equations to remember:
∆d = df – di or ∆v = vf – vi
 Therefore, you add the second vector to the
opposite of the first vector.

20. Subtraction of Vectors (cont.)
 Ex. = Vector #1: 5 km East
Vector #2: 4 km North
5 km W (-v1)
4 km N (v2)
I know it seems silly, but trust me on
this one!!!

21. Component Method of Vector Addition
 Treat each vector separately:
1. To find the “X” component, you must:
Ax = Acos Θ
2. To find the “Y” component, you must:
Ay = Asin Θ
3. Repeat steps 2 & 3 for all vectors

22. Component Method (cont.)
4. Add all the “X” components (Rx)
5. Add all the “Y” components (Ry)
6. The magnitude of the Resultant Vector is
found by using Rx, Ry & the Pythagorean
Theorem:
RV2 = Rx2 + Ry2
7. To find direction: Tan Θ = Ry / Rx

23. Component Method (cont.)
Ex. #1
V1 = 2 m/s 30° N of E
V2 = 3 m/s 40° N of W
(this is easy!)
Find: Magnitude & Direction
Magnitude = 2.96 m/s
Direction = 78° N of W

24. Component Method (cont.)
Ex. #2
F1 = 37N 54° N of E
F2 = 50N 18° N of W
F3 = 67 N 4° W of S
(whoa, this is not so easy!)
Find: Magnitude & Direction
Magnitude =37.3 N
Direction = 35° S of W