1. Vectors Vector vs Scalar Quantities and Examples
Vector Addition – Graphical and Analytical Methods

2. What is a Scalar Quantity?
A scalar quantity has magnitude (amount, includes a number and unit) only.
Some examples of scalar quantities are:
distance (5 m)
speed (20 m/s)
mass (3 kg)
time (4 sec)
volume (30 ml)

3. What is a Vector Quantity?

4. Drawing a Vector
Choose an appropriate scale for the vector. Make sure the scale doesn’t make the vector less than 2 cm long.
Using graph paper, mark your origin (starting point). If no graph paper, make an origin like this:
Measure the angle of the vector from east (or 0° or positive x-axis) using a protractor and mark it. Draw a light line from the origin to that mark.
If the angle is beyond 180°, turn your protractor upside down.

5. Drawing a vector con’t Example: Draw 40 km @ 150° Scale: 1 cm = 10 cm
Starting at the origin, darken the line for the correct length according to the scale and include an arrow tip to indicate the direction.
Example: Draw °
Scale: 1 cm = 10 cm
4 cm
150°

6. Distance vs Displacement
Distance – length, “how far”, scalar
Displacement – length and direction, “how far” in a given direction, vector, final position – initial position
Example:
Total distance =
2m + 3m + 2m = 7m
Displacement = 3 m, east
3 m, E
2 m, N
2 m, S
3 m, E
displacement

7. Vector Addition
When two or more vectors are added, the directions must be considered.
Vectors may be added Graphically or Analytically.
Graphical Vector Addition requires the use of rulers and protractors to make scale drawings of vectors tip-to-tail. (Less accurate)
Analytical Vector Addition is a mathematical method using trigonometric functions (sin, cos, tan) and Pythagorean theorem.

8. Resultant The resultant is the sum of the vectors being added.
The resultant vector is drawn from the tail of the first vector (origin or starting point) to the tip of the last vector (end or finish).
The angle for the resultant is measured using a protractor at the origin or starting point. If possible measure from east (0° or positive x-axis) and that is the resultant angle, since angles are assumed to be measured from east counter-clockwise.

9. Resultant – con’t
If not possible, measure the acute angle from N (90°), W (180°), S (270°), or (360°).
Adjust it by adding directions such as N of W.
Or by adding (or subtracting) the acute angle to (from) 90°, 180°, 270°, or 360° depending on whether the acute angle is before(subtract) or after (add) 90°, 180°, 270°, or 360°.
Ex: The angle below would be 30° S of W or 210° (180° + 30° since the 30°
is past 180°).
30°

10. Graphical Addition of Several Vectors-
Draw each of the vectors tip-to-tail to scale.
Draw the resultant from the tail of first vector to tip of last vector (start to finish)
Given vectors A, B, and C C B R
R = A + B + C A
 (direction of R)

11. Graphical Addition - Resultant
R = A + B + C A B C
 (direction of R)
Given vectors A,B,&C R
To get the magnitude of the resultant, measure the length of R and multiply by the scale factor.
To get the direction of the resultant, measure the angle from 0° (or E or +x) to the resultant vector. If more than 180°, see the previous resultant slide.

12. Vector Addition – 1D
When adding vectors in the same direction, add their magnitudes and the resultant direction is the same.
Ex: 5 m, E + 10 m, E = 15 m, E
When adding vectors in the opposite direction, subtract their magnitudes and the resultant is in the same direction as larger magnitude.
Ex: 5 m, E + 10 m, W = 5 m, w

13. Right Triangle Trigonometry
sin  = opposite hypotenuse cos  = adjacent hypotenuse tan  = opposite adjacent
hypotenuse
opposite C B  A
adjacent
And don’t forget: Pythagorean Theorem A2 + B2 = C2

14. Adding 2-D vectors that form a right triangle analytically
Sketch the vectors tip to tail. (no scale needed)
Calculate the magnitude of the resultant using the Pythagorean theorem.
Calculate the angle using inverse tangent. R
18 m
24 m

15. Stating the final answer
All vectors must be stated with a magnitude and direction.
Angles must be adjusted by adding compass directions ( i.e. N of E) or angle adjusted to be measured from East (or 0° or the positive x-axis).
When added graphically, the length of the resultant must be multiplied by the scale to find the magnitude.