1. Vectors

2. The Basics: the notes How to add vectors
Vectors should be added from “tip/head to tail”
For example
4m west + 3m south would be
4 meters west
3 meters south

3. 4 meters west
3 meters south
resultant
Tip of final arrow
Starting point
The answer or resultant will have both a magnitude (number) and a direction.
The resultant (usually dashed) should be drawn from the starting point to the tip of the last arrow head
ALWAYS INCLUDE AN ARROWHEAD ON YOUR RESULTANT!
At this point, the magnitude and direction of the answer/resultant could be calculated…

4. Finding the resultant’s magnitude and direction

5. Finding the Magnitude (5 m) C = 5m C = A2 + B2
When using vectors there will always be triangles so Use the Pythagorean theorem
A2 + B2 = C2 (where C is the hypotenuse)
C = 5m
C = 3m m2
C = A2 + B2
4 m
3 m
(5 m)
We have found the magnitude of the vector
Resultant=

6. Finding the direction Sin θ = opposite/hypotenuse
Use trigonometry to find the angle at the base of the resultant vector
Call this angle theta (θ)
Sin θ = opposite/hypotenuse
Cos θ = adjacent/hypotenuse
Tan θ = opposite/adjacent
Adjacent side θ
4 m
Opposite side
3 m
hypotenuse
(5m)
base

7. Finding the direction θ (5m) 4 m base Adjacent side Opposite side 3 m
hypotenuse
(5m)
base

8. Finding the direction cont.
When choosing which sides to use…
choose the original givens not the side you calculated
In this case the 3 meters and 4 meter side
Which function should you use?
Sin θ = opp/hyp
Cos θ = adj/hyp
Tan θ = opp/adj
Adjacent side θ
Tan θ = opp/adj
4 meters west
Opposite side
3 meters south
(5m)

9. Finding the direction cont.
Tan θ = opp/adj
Tan θ = 3m/4m
You will need to use the inverse tan (2nd function- then push tan) on the cal.
Cal=
R= θ = 400
Adjacent side
4 meters west θ
400
Opposite side
3 meters south 5m

10. Finding direction cont
Direction will be given in degrees from 0 (due east)
900
1800
00= due east
2700

11. Place the resultant vector on a coordinate plane (the base will always be at the origin)
900
1800
00= due east
Base of resultant vector
5 m
2700

12. 5 m Final answer= 180 + 40 Final answer= 2200
We also know the angle we just calculated
Now determine how far from 00 east this angle is…
Final answer=
Final answer= 2200
We have now found the direction of the resultant vector!
900
1800
00= due east
400
Base of resultant vector
5 m
2700

13. Basic 2 Vector problem: [Two Forces act on an object.
One is a northern force of 24.0 N. The other is an easterly force of 13.0 Newtons.
What is the resultant force magnitude and direction on the object overall?]

14. *Final answer must be from 00 (due east) Resultant magnitude:
(opp of θ)
Resultant direction:
13.0 N
900
Tan θ = opp/adj
Tan θ= 13.0N/24.0 N
24.0N
Cal=
(adj of θ)
(27.3 N)
clip..\.OB
θ will be the angle at the base of the resultant
28.40
R= 28.40 θ ?
61.60
*Final answer must be from 00 (due east)
Resultant magnitude:
A2 + B2 = C2
C = A2 + B2
90.0 – 28.4= 61.60
C = N N2
R= 27.3 N
Cal =

15. One dog… three leashes However, vectors must be added head to tail
This shows the reality of the situation…

16. 2.5 N
The order added doesn’t matter- the resultant is the same!
5.5 N N
Up and left
And also…
1 leash is enough!
4.0 N

17. Rem., order does not matter

18. Bellwork: Date:
Two students pull on a wishbone. One pulls at 34.9 N west, the other at 13.3 N south.
1) Draw the reality of the situation with labels. (not tip to tail- but how it would actually look from above)
2) What is the resultant magnitude?
Must re-draw vectors tip to tail
3) What is the resultant direction?
37.3 N
( =)
(one past decimal- NOT sig. figs.!)

19. Component Vectors

20. Vector Components
Consider a 24 N force at 30.00
24 N
30.00

21. Vector Components To the right And upward
What two directions does this arrow point?
To the right
And upward
We say that this vector has both a Easterly component and a Northerly component
24 N
30.00

22. Vector Components cont.
Calculate how much of the force of this vector acts in the northerly direction?
Note that you have a side (24N) and an angle(30.00).
Note also that the northern component is OPPOSITE the angle given.
We also know the HYPOTENUSE (24N)
Which function will be used?
Sin θ = Opp/Hyp
Sin 30.0 = opp/ 24N
Cal= 12
R= 12 N northward
24 N
(12N)
30.00

23. Vector Components cont.
Calculate how much of the force of this vector acts in the Easterly direction.`
Note also that the eastern component is ADJACENT to the angle given.
HYPOTENUSE (24N)
Which function will be used?
Cos θ = Adj/Hyp
Cos 30.0 = adj/ 24N
Cal=
R= 21 N eastward
24 N
30.00
(21N)
Vector components clip..\..\

24. Bellwork (pre-mini warm-up): Basic 2 vector and Component vectors
This represents a boat and a current:
A) 25.9 m/s east B) 33.3 m/s south
1) What is the resultant magnitude?
2) What is the resultant direction?
3) If a car is traveling 11.1 m/s at , how fast is it moving eastward? (Use the 10.0 not the 80.0 degree triangle)
42.2 m/s
52.10
= 307.9
1.93 m/s east