1. Vectors in Two Dimensions
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2. Scalars and Vectors Scalar quantities have only magnitude.
All measurements are considered to be quantities. In physics, there are 2 types of quantities – SCALARS AND VECTORS.
Scalar quantities have only magnitude.
Vector quantities have magnitude and direction.
time
mass
temperature
displacement
Magnetic Field
acceleration
Force
velocity
Gravitational Field

3. Vectors are used to describe motion and solve problems concerning motion.
For this reason, it is critical that you have an understanding of how to
represent vectors
add vectors
subtract vectors
manipulate vector quantities.
in 2 and 3 dimensions.

4. Vectors tip 8 units 2 units 5 units tail
Magnitude represented by the length of the vector

5. Vectors y
1200
800 00
2250
300
600 from -x
3000 x
-600
450
from-y
Direction represented by the direction of the arrow

6. Adding Vectors - + Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi
We know how to add vectors in 1-dimension.
Example: displacement. If someone walks 4 mi east (Dx1) and then 7 mi west (Dx2) the total displacement (Dx1 + Dx2) is 3 mi west.
1. Adding vectors GRAPHICALLY – place them TAIL TO TIP
Dx1 + Dx2 = 3 mi
Dx1 = 4 mi
west -
east +
Dx2 = 7 mi
2. Adding vectors MATHEMATICALLY – In 1- dimension, assign direction + or – and add algebraically
Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi

7. Adding Vectors
What about if the vectors are in different directions in 2-D?
How do we describe the direction?
How do we add/subtract the vectors?
v1 = 5
35o below +x
v2 = 3
50o above +x

8. Adding Vectors Graphically
“tail to tip”
To add vectors using the tail to tip method
Draw the first vector (7u, 50o N of E) beginning at the origin.
Draw the second vector (3u, 35o S of E) with its tail at the tip of the first vector.
Draw the Resultant vector (the answer) from the tail of the first vector to the tip of the last.
North
South
East
West v2 v1
v = v1 + v2

9. Adding Vectors Grahically “parallelogram method”
To add vectors with the parallelogram method
Draw the first vector to scale beginning at the origin.
Draw the second vector, to scale, with its tail also at the origin.
Starting at the tip of one vector, draw a dotted line parallel to the other vector. Repeat, starting from the tip of the second vector.
Draw the Resultant vector (the answer) from the origin to the intersection of the dotted lines.
North
South
East
West v1
v = v1 + v2 v2

10. Adding Vectors Mathematically
North
South
East
West
What if the vectors are in different directions?
For example, what if I walk 5 steps north and then 4 steps east. What is my total displacement , Dx, for the trip?
OR what is the vector sum of the Dx1 and Dx2?
Dx2 =4 steps east
Dx1=
5 steps north
Dx = Dx1+Dx2 = ?

11. Adding Vectors Mathematically
4 steps east
5 steps north
Dx = ? q
Use Pythagorean Theorem to find the magnitude of Dx
Use right triangle trig to find the direction of Dx

12. RIGHT TRIANGLE TRIGONOMETRY
A – side adjacent to angle q
O – side opposite to angle q
H – hypotenuse of triangle
Pythagoreon Theorem
SOHCAHTOA H
sinq = O H O
cosq = A H q
tanq =
sinq
cosq = O A A

13. R = 283 m R2 = A2 +B2 = 2402 + 1502 tanq = opp adj 150m 240m =
A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?
North
MAGNITUDE
R2 = A2 +B2 =
R = 283 m
DIRECTION
A = 240 m
West
East
tanq =
opp
adj
150m
240m = q B=
150m A
R=283 m
tanq = 0.625
= tan-1(1.3) = 32o south of east
Notice that A and B are perpendicular components of R. They are the amount of R in each direction
South

14. The net displacement is 283 m in the direction of 32o S of E.
A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?
North
The net displacement is 283 m in the direction of 32o S of E.
The average velocity:
240 m
West
East q
v = Dx Dt
283 m
0.5 hr
150m =
283 m
= 566 m/hr in the direction
32o S of E.
South

15. R = 9.2 u R2 = A2 +B2 = 82 + 4.52 tanq = opp adj 4.5 u 8 u =
EXAMPLE: What is the Resultant of adding 2 vectors, A and B, if A = 8 units south and B = 4.5 u west?
MAGNITUDE
R2 = A2 +B2 =
North
R = 9.2 u
DIRECTION
tanq =
opp
adj
4.5 u
8 u
West
East =
tanq = 0.56
q-1 = tan-1(0.56) = 29o
9.2 u
8u south R q
The Resultant is 9.2 units in the direction of 29o south of west
South
4.5u west
Notice that A and B are perpendicular components of R. They are the amount of R in each direction

16. Perpendicular Components of a Vector
Any vector can be resolved into perpendicular components. Use right triangle trig – x and y components always make a right triangle with the vector .
Ex. Vector A has magnitude 8.0 m at an angle of 30 degrees below the x-axis. What are the x- and y-components of A?
cos30 = Ax A
Ax = Acos30 Ax
Ax = 8cos30 = 8(0.866) = +6.9
30o Ay
A=8
sin30 = Ay A
Ay = Asin30
Ay = 8sin30 = 8(0.5) = - 4m
You must assign the correct direction

17. x- and y- Components of a Vector
What are the x- and y- components of the vector A, shown below? y
A = 8 m Ay
Ax = Acosq
= 8cos(30)
= 8(0.866)
= -6.9 m
30o x Ax
Ay = Asinq
= 8sin(30)
= 8(0.5)
= +4 m
Ax is the amount of A in the x-direction
Ay is the amount of A in the y-direction

18. A butterfly moves with a speed of 12 m/s
A butterfly moves with a speed of 12 m/s. The x-component of its velocity is 8.00 m/s. The angle between the direction of its motion and the x-axis must be what?
North
v = 12 m/s vy
q ?
West
cosq = vx v 8 12 =
vx = 8
East
q = cos = 48o
South

19. Adding Vectors by Components
What about adding 2 vectors, A and B, that are NOT perpendicular or parallel?
A + B = R Bx y
(Ax+Ay)+(Bx+By)=(Rx+Ry) By B
Any vector can be described as the sum of perpendicular components.
60o
Components in the same direction are added as 1-D vectors to find the components of the resultant vector. Ry R A
Rx = Ax + Bx
= 0 + Bx x Rx
Ry = Ay + By

20. Adding x- and y- Components of a Vector
A + B = R
(Ax+Ay)+(Bx+By)=(Rx+Ry) y Bx
Determine the perpendicular components of each vector. Make a table to add up x and y components separately: By B
60o x Y A Ax Ay B Bx By R Rx Ry R Ry A q Rx x
Magnitude of R:
R2 = Rx2 + Ry2
Direction of R:

21. ADDING VECTORS GRAPHICALLY You can add as many vectors as you want
North
A + B + C + D = R R D
Graphically, the vectors are added “tail to tip”
East
West A
and the order doesn’t matter C B
C + A + D+ B = R
South

22. A + B + C + D = R D A C B Mathematically ADDING VECTORS
(Ax+Ay)+(Bx+By)+(Cx+Cy)+(Dx+Dy)=(Rx+Ry)
To Add Vectors by Components:
Place each vector at origin. Find the x- and y- components of each vector in the sum, and list them in a table. Make sure to include the direction of each component by hand.
Add all the x-components to find Rx.
Add all the y-components to find Ry.
Use Pythagorean Theorem and trigonometry to find the magnitude and direction of R. x Y A
-Ax
-Ay B
+Bx
-By C
+Cx
+Cy D
-Dx
+Dy R Rx Ry D A
R2 = Rx2 + Ry2
Magnitude of R: C
Direction of R: B

23. EXAMPLE (Adding Vectors by Components)
Determine the resultant of the following 3 displacements:
A. 24m, 30º north of east
B. 28m, 37º east of north
C. 20m, 50º west of south Bx
North B A
x (m)
y (m) A
20.8
12.0 B
16.9
22.4 C
-15.3
-12.9 Ʃ
21.5 By
37o Ay
West
30o Ax
East
50o Cy C Cx
South
South

24. R = 31 m, 440 N of E (from the x-axis)
EXAMPLE R
North
x (m)
y (m) A
20.8
12.0 B
16.9
22.4 C
-15.3
-12.9
Ʃ R
21.5 Ry q
West Rx
East
South
Magnitude
Direction
R = 31 m, 440 N of E (from the x-axis)
South

25. DO NOW
An airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?
x (km)
y (km) A
620 B
311
-311 C
-331
-439 DR
600
-750
North A Cx
West Bx
East
45o By
53o Cy B C
South DR

26. DR = 960 km, -510 from the x-axis (510 S of E)
EXAMPLE (cont.)
North
x (km)
y (km) A
620 B
311
-311 C
-331
-439 DR
600
-750
West
East q DR
South
Magnitude
Direction
DR = 960 km, -510 from the x-axis (510 S of E)

27. DR EXAMPLE q 960 km at 510 South of East
An airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?
North
West
East q DR
South
960 km at 510 South of East

28. - + = = - -v2 -v2 Subtracting Vectors v1 v2 v1 v1 v2 v1
In order to subtract a vector, we add the negative of that vector.
The negative of a vector is defined as a vector in the OPPOSITE direction (with each component the negative of the original) v1 v2 - v1
-v2 + =
-v2 = v1 v2 - v1

29. Graphical Representation
B=10
60o -B
60o
A + B = S
A – B = D -B A D
Tail to tip
Tail to tip
(A and –B) A B A B S
Tail to tail
(A and B) D

30. - 5 + 8.7 Mathematical Representation A=8 B=10 A + B = S A + (–B) = Dx y A
+ 8 B
+ 5
- 8.7 S 13
-8.7 x y A
+ 8 -B
- 5
+ 8.7 D 3
+8.7 -B
60o S Sx Sy q D Dx Dy q

31. Scalar Multiplication
Multiplication of a vector by a positive scalar changes the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar "scales" the vector.
Multiplication of a vector by a negative scalar changes the magnitude of the vector, and makes the direction opposite.
Example:
3A = 3 x = =
If the scalar is negative, it changes the direction of the vector.
-3A = -3 x = =