1. 4.1 Vectors in Physics
Objective: Students will know how to resolve 2-Dimensional Vectors from the Magnitude and Direction of a Vector into their Components/Parts and, Calculate the Magnitude and Directions of a Vector from its Components or Parts.
2 Dimensional
Vector with x
and y
Component
/parts
1 Dimensional
Vector in positive x direction.
y part
x part
1 Dimensional
Vector in the negative
y direction.
Notice direction of Component Vectors !!!

2. A Vector is like an Arrow that has a Magnitude and a Direction
A Vector is like an Arrow that has a Magnitude and a Direction. The Direction is specified by and Angle or it can be represented by Components. D
= o or D
= 3 x + 4 y
Vectors are always straight lines. This means that Vectors have 1 Direction
The Magnitude is the Size or Length of the Vector and can be easily represented/drawn using a Ruler and Graph Paper.
The Direction is either specified by both a Magnitude and Angle or in Component Form using x and y to designate each sub-vector / sub-part.
When drawing Vectors on Graph Paper represented by a Magnitude and Angle a Protractor and Ruler needed.
Do you draw and measure a Vectors Magnitude with a Ruler or with a Protractor ?
In our Class so far Distance, Time, Average Speed have been designated as Scalar and Displacement, Velocity and Acceleration are Vectors.

3. 1) For each of the following quantities indicate whether it’s a Scaler or a Vector ?
a) The time it takes you to run 1 mile.
Scalar
b) Your Displacement after running 100 meters.
Vector
c) Your Average Velocity while running a 10 K.
Vector
d) The Speed read on your odometer while driving to LA.
Scalar
2) Rank the Vectors in order of increasing Magnitude ! y x B C D A
C < B < A < D

4. A Vector is a Quantity with a Magnitude and a Direction.
We said “Magnitude is the Size or Length of the Vector”.
How do we write this Vector with a Magnitude and a Direction ?
The Magnitude is calculated using the Pythagorean Theorem, by squaring the components of the Vector. |V| = (x2 + y2)
What is the Value of the X component of this Vector when the Magnitude is calculated below ?
V = 3
270o
| V | = (-3)2 = = 9 = 3
V =
-3y
Similar to | V | = | - 3 | = 3 but, this is a short-cut for 1 dimensional Vector.

5. A Vector is a Quantity with a Magnitude and a Direction.
The Direction is given by the Angle of the Vector, or determined when a Vector is written in Component Form and then Graphed.
We specify Vector Components / Directions by using x, –x, +y, –y
90o or +y
0o (360o) or +x
270o South or –y
180o or –x
+53o 5
How did we get from the a Magnitude and Angle to its Components ? D
= o or D
= 3 x + 4 y

6. Lets Find components from Magnitude and Angle (Direction) :5 y
= 5 sin 53o = 4 y x
= 5 cos 53o= 3 x
Adj = x
Opp = y
The Vectors component’s give each an every Vector it’s own unique coordinate system !!! D
= 3 x + 4 y
That’s what make them SPECIAL, they can be moved around and they still have their components.

7. = ( opposite / adjacent ) = ( y / x )
Now Let go back the other way from Component Form to Magnitude and Direction !
So the Direction from the above Vector Components is obtain from the definition of the tangent angle of a Right Triangle: D
= 3 x + 4 y
Tangent angle = tan
= ( opposite / adjacent ) = ( y / x )
Therefore, the angle Theta is the arctan or inverse tangent of the opposite over adjacent or y / x . y x 3
= tan -1 ( 4 )
) = 53o
And again the Magnitude is the square root of the sum of the components squared or the Pythagorean Theorem :
| D | = (4)2 = = 25 = 5 D
= o

8. 1) Why is a Velocity of -30 m/sec x (30 West) greater than a Velocity 20 m/sec x (20 East) ?
Because its Magnitude is 30 versus 20. Squaring -30 and taking the Square Root of 900 equals 30.
| V | = (0)2 = = 30
2 ) Find the components from the Magnitude and Angle of : D
= 20 < 30o
= 30o 20 x
= 20 cos 30o= 20 3 2
x = 17.1 x y
= 20 sin 30o= 20 1 2
x = 10 y D
= 17.1 x + 10 y
3 ) And What is the inverse Tangent of ( 10 / 17.1 ) y x
17.1
= tan -1 ( 10 )
) =
30o

9. If a Vector is in 1- Dimension:
We specify direction by using x, –x, +y, –y
Any Vector 1 or 2 – Dimensional, can be moved anywhere on their own unique axis !!! They are movable entities !!
–3y
+5y
+5y
There location does not matter ! Vectors give you FREEDOM !!!
–6x
+2x
–3y
+2x
Can I combine an x component of Vector A to a y component of Vector B ?
NO !
Can I combine an x component of Vector A to an x component of Vector B ?
YES !

10. Vectors at Angles 90o North +45o 0o East (360o) 180o West 270o South
When Vectors are not aligned to an axis, direction is given as an angle measured counterclockwise from the +x axis.
0o East (360o)
90o North
270o South
180o West
+45o

11. Vectors at Angles 90o North +135o 0o East (360o) 180o West 270o South
When Vectors are not aligned to an axis direction is given as an angle measured counterclockwise from the +x axis
0o East (360o)
90o North
270o South
180o West
+135o

12. Vectors at Angles 90o North +225o 0o East (360o) 180o West 270o South
When Vectors are not aligned to an axis direction is given as an angle measured counterclockwise from the +x axis
0o East (360o)
90o North
270o South
180o West
+225o

13. Vectors at Angles 90o North 0o East (360o) 180o West +315o 270o South
When Vectors are not aligned to an axis direction is given as an angle measured counterclockwise from the +x axis
0o East (360o)
90o North
270o South
180o West
+315o

14. Negative Angles 90o North 0o East (360o) 180o West –45o +315o
You can measure angles clockwise but, you must specify them as negative angles.
0o East (360o)
90o North
270o South
180o West
–45o
+315o

15. Adding Vectors Vectors can be added to one another.
Only like vectors can be added to each other.
Displacement adds to displacement
Velocity to velocity, etc.
You CANNOT add displacement to velocity.
However, when you multiply the velocity vector by the scalar time it becomes a displacement vector, which can be added to the objects initial displacement vector. x = x0 + vt .
When you multiply the acceleration vector by the scalar time it becomes a velocity vector, which can be added to the objects initial velocity vector. v = v0 + at .
Adding some Vectors is very conceptual, so lets start with something that’s easy to visualize.
Adding ordinary displacement vectors.

16. 1) An object follows the path in the diagram below
1) An object follows the path in the diagram below. At the end of the motion what is the total Distance traveled and the Magnitude and Direction of the Displacement Vector?
-3 Km
Distance = =
D = 4.72
4 Km
Distance = = 12 Km
63.4o D
= 5 Km x Km x + 4 Km y =
5 Km
= 2 Km x + 4 Km y D
| D | = (4)2 = = 20 = Km y x 2
= tan -1 ( 4 )
) = 63.4o
= o D

17. There are 2 Methods to Add Vectors !
Tip to Tail Method.
By putting the Tip of one Vector you want to ADD on Graph Paper to the Tail of the other Vector to be added and drawing the Result from the Tail of the 1st Vector to the Head of the 2nd Vector.
Using Graph Paper, Ruler and a Protractor makes this easy.
2) Vectors can be Added with Components.
By Adding the x components of 1 Vector to the x component of the 2nd Vector and then adding the y component of 1 Vector to the y component of the 2nd Vector.
Similar to adding like terms in Algebra !!!

18. Method 1 -Tip-to-Tail Vector Addition
One visual method of vector addition is Tip-to-tail.
The first vector is drawn, setting the scale for all subsequent vectors. (A vector of 100 should be twice as long as a vector of 50, etc.)
The tail of the next vector is added to the tip of the first vector.
This continues until all vectors are added tip-to-tail.
Then walk 25 m at 90o
Walk 50 m at 45o
Walk 100 m at 0o
Start here

19. Resultant Final This is called the resultant R , or the vector sum
The resultant is the result (sum) of adding vectors.
The resultant is a vector pointing from initial to final. For displacement vectors it is the shortest straight-line displacement from initial position to final position.
Note: The resultant is NOT drawn tip-to-tail.
Final
This is called the resultant R , or the vector sum
Initial

20. Vector Addition
The red resultant vector does the same job as the three black vectors added together.
The resultant is mathematically equal to the sum of vectors. C B R A
Clearly, adding vectors is not just a matter of adding their magnitudes.
1) This type of addition will require Geometry and Trigonometry.
2) Or You could use Graph Paper, a Ruler and a Protractor

21. Adding Vectors on an Axis
Example: Moving 20 m, +x direction followed by 10 m, –x direction.
The object has a final (resultant) displacement of 10 m, +x direction.
Vectors on the same axis can be added using ordinary addition & subtraction.
Direction is easy. or
= o R or
= o R
+20m
R = +10 m
–10m

22. What if they are on different axes ?
Example: 20 m, +x , then 10 m, +y.
Perpendicular vectors can ALWAYS be added with Pythagorean Theorem to find the resultants magnitude.
Trig is needed to find the resultants direction.
R = ?
θ = ?
+10m
22.4
26.6
+20m

23. What if they are both at angles ?
Example: 20 m at 45o, then 10 m at 30o.
+20m
45o
+10m
30o

24. +10m 30o +20m 45o Draw the 1st vector at the origin.
At the tip of the first vector imagine a new coordinate axis.
Then add the next vector tip-to-tail using the imagined second axis as a guide. Keep doing this until all vectors are added.
+10m
30o
+20m
45o

25. How do we find the resultant ?????
Remember the resultant is not tip to tail, it is initial to final.
+10m
30o
+20m
45o
R= ?
Now use your Ruler to measure the Magnitude and your Protractor to find the Angle of the Resultant Vector R

26. Adding Vectors on an axis was easy.
Method Adding Vectors with Components.
Adding Vectors on an axis was easy.
Now lets convert these vectors to vectors that lie along axes, which are easier to add ? We will add like components to like components !
+10m
30o
+20m
45o

27. Split these vectors into components
Always Draw the x-component first.
Then tip to tail the y-component vector to the x-components.
+10m
30o
+20m
45o

28. Use Cosine and Sine y y x +10m 30o +20m 45o
x component = Hypotenuse ( Cos Angle )
y component = Hypotenuse ( Sin Angle ) x y
+10m
30o
+20m
45o y

29. x y
+10m
30o
+20m
45o y

30. y
+10m
30o
+20m
45o
8.66 y
14.1

31. +10m
30o
+20m
45o y y
8.66
14.1

32. +10m
30o
+20m
45o
5.00
8.66
14.1
14.1

33. Do the component vectors ( x and y ) added together get to the same location as the two original vectors ?
+10m
30o
+20m
45o
14.1
8.66
5.00

34. Yes they do!
And they can be added together the easy way.
14.1
8.66
5.00

35. Group the x-components on the x-axis and y-components on the y-axis
14.1
8.66
5.00

36. Add the x-component vectors to get the resultant in the x-direction Rx .
Add the y-component vectors to get the resultant in the y-direction Ry .
14.1
5.00
14.1
8.66

37. Now you have two vectors that are perpendicular
22.8
19.1

38. After summing the components, draw the resultant.
Is this the resultant we want?
Check the original vectors
Yes, it is. It gets to the same place as the original vectors.
+20m
45o
+10m
30o
R = ?
θ = ?
22.8
19.1

39. Magnitude and direction of the resultant
θ = ?
22.8
19.1
In first quadrant the reference angle equals the angle from the x-axis
40 degrees

40. Mathematically 1. Find components
2. Sum the x components, and sum the y components.
3. Pythagorean theorem to find the magnitude of R .
4. Inverse tangent to find the direction of R .
This is a first quadrant vector, so the reference angle is also the vector angle measured from the x-axis.

41. Resultant Direction can be Tricky
The previous example finished with a resultant in the 1st quadrant.
However, if the resultant is in any other quadrant you will need to refer to the drawing of Rx, Ry, and R to adjust the final angle to east.
Example: If Rx = –30m and Ry = +40m
The inverse tangent formula often calculates a reference angle, in this case 53o.
When drawn tip-to-tail, as shown at the right, it is apparent that this is reference angle. (Draw the x component first, then draw the y component tip-to-tail.)
Now it is easy to adjust the angle so it is measured from the +x axis.
θ = 180o − 53o = 127o
+40m
– 30m R
127o
53o

42. Example: 30m at 150o, then 50m at 240o
Sketch of the problem.
50 m
30 m
150o
240o

43. 1. Find components
50 m
30 m
150o
240o
–25.0
–43.3
+15.0
–26.0

44. 2. Sum the x components, and sum the y components.
–25.0
–43.3
+15.0
–26.0
–28.3
–51.0

45. 3. Pythagorean theorem to find resultant magnitude
–28.3
–51.0
R = 58.3 m

46. 4. Inverse tangent to find reference angle
Remember to adjust angle
209o θ
–28.3
–51.0
R = 58.3 m

47. 209o
R = 58.3 m
209o = x – 28.3 y
R = 58.3 m
R = 58.3 m
209o
R = x – 28.3 y

48. That’s It !!! 29o -R = 58.3 m R = 58.3 m 209o 209o R = - 51 x – 28.3 y
Think and tell me how would you subtract 2 Vectors ? Relate it to Subtracting like terms in Algebra ! What do you do to the term you want to subtract ?
When you Subtract Vectors, you subtract components. Take the inverse or opposite sign of each component of the Vector being Subtracted. Also, when graphing subtraction the negative Vector points in the opposite direction. That is, multiplying a Vector by -1 reverses its direction. Once the components are found the Magnitude and direction Angle is calculated as before.
-R = 58.3 m
29o
R = 58.3 m
209o
209o
R = x – 28.3 y
29o
-R = x y
That’s It !!!

49. Summarize in your Notebook/Journal
Summarize in your Notebook/Journal. Write how to find the Magnitude of a Vector, the Angle of a Vector and, and describe how to Add 2 Vectors.
Write formulas for calculating Magnitude, calculating Angle, calculating the Components of Vectors and, describe how to Add 2 Vectors !!!
Homework: Page Lesson Check and, Page Lesson Check !!! 3 2 3 2 1 2