1. Chapter 3
Vectors

2. Vectors and Scalars,
Addition of vectors
Subtraction of vectors

3. Magnitude Direction VECTORS !!!!!
Physics deals with many quantities that have both
Magnitude
Direction
VECTORS !!!!! y  x r

4. Scalar Examples of Scalar Quantities: Length Area Volume Time Mass
A scalar quantity is a quantity that has magnitude only and has no direction in space
Examples of Scalar Quantities:
Length
Area
Volume
Time
Mass

5. Vector
A vector quantity is a quantity that has both magnitude and a direction in space
Examples of Vector Quantities:
Displacement
Velocity
Acceleration
Force

6. A vector is a quantity that has both magnitude and direction
A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.
Blue and green vectors have same direction but different magnitude.
Blue and purple vectors have same magnitude and direction so they are equal.
Blue and orange vectors have same magnitude but different direction.
Two vectors are equal if they have the same direction and magnitude (length).

7. Examples
A = 20 m/s at 35° NE
B = 120 lb at 60° SE
C = 5.8 mph/s west

8. Example The direction of the vector is 55° North of East
The magnitude of the vector is 2.3.

9. Try Again
Direction:
Magnitude:
43° East of South 3

10. Try Again
It is also possible to describe this vector's direction as 47 South of East.
Why?

11. Vector Addition Triangle (Head-to-Tail) Method
vectors may be added graphically or analytically
Triangle (Head-to-Tail) Method
1. Draw the first vector with the proper length
and orientation.
2. Draw the second vector with the proper length
and orientation originating from the head of
the first vector.
3. The resultant vector is the vector originating
at the tail of the first vector and terminating
at the head of the second vector.
4. Measure the length and orientation angle of
the resultant.

12. Adding vectors in same direction:
Example: Travel 8 km East on day 1, 6 km East on day 2.
Displacement = 8 km + 6 km = 14 km East
Example: Travel 8 km East on day 1, 6 km West on day 2.
Displacement = 8 km - 6 km = 2 km East
“Resultant” = Displacement

14. Adding more than two vectors graphically

15. Subtraction of Vectors -graphically

16. Parallelogram (Tail-to-Tail) Method
1. Draw both vectors with proper length and
orientation originating from the same point.
2. Complete a parallelogram using the two
vectors as two of the sides.
3. Draw the resultant vector as the diagonal
originating from the tails.
4. Measure the length and angle of the
resultant vector.

17. Components of Force: y x

18. Resolving a Vector Into Components+y
The horizontal, or
x-component, of A is
found by Ax = A cos q. A Ay q Ax
The vertical, or
y-component, of A is found by Ay = A sin q +x
By the Pythagorean Theorem, Ax2 + Ay2 = A2
Every vector can be resolved using these
formulas, such that A is the magnitude of A, and
q is the angle the vector makes with the x-axis.

19. Rx2 + Ry2 = R2 Analytical Method of Vector Addition
1. Find the x- and y-components of each vector.
Ax = A cos q
Ay = A sin q
Bx = B cos q
By = B sin q
Cx = C cos q
Cy = C sin q Rx Ry
2. Sum the x-components.
This is the x-component of the resultant.
3. Sum the y-components.
This is the y-component of the resultant.
4. Use the Pythagorean Theorem to find the
magnitude of the resultant vector.
Rx2 + Ry2 = R2

20. q = Tan-1 Ry/Rx 5. Find the reference angle by taking the inverse
tangent of the absolute value of the y-component
divided by the x-component.
q = Tan-1 Ry/Rx
6. Use the “signs” of Rx and Ry to determine the
quadrant. NW NE
(-,+)
(+,+)
(-,-)
(-,+) SW SE

21. Step 1
Sketch the given vector with the tail located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º)
25 m
36º

22. Step 2
Draw a line segment from the tip of the vector perpendicular to the x-axis
25 m
36º
Notice, you now have a right triangle with a known hypotenuse and known angle measurements

23. Step 3
Replace the perpendicular sides of the right triangle with vectors drawn tip – to - tail
25 m

24. Step 4
Use sine and cosine functions to find the horizontal and vertical components of the given vector.
25 m Ry
36º Rx
Cos(36) = Rx/25
Rx = 25cos(36)
Rx = 20.2 m
Sin(36) = Ry/25
Ry = 25sin(36)
Ry = 14.7 m

25. Example: x y + 0.09 + 6.74 R R = (0.09)2 + (6.74)2 = 6.74 N
5 cos 30° = +4.33
5 sin 30° = +2.5
6 cos 45 ° =
6 sin 45 ° =
+ 0.09
+ 6.74
6 N
5 N
135°
45°
30° Ry R
R = (0.09)2 + (6.74)2
= 6.74 N Rx
 = arctan 6.74/0.09
= 89.2°

26. Problem 1

27. Problem 2

28. Problem 3

29. Problem 4