1. Kinematics & Dynamics in 2 & 3 Dimensions; Vectors
Review of some
Math Topics
in Ch. 1.
Then, some
Physics Topics
in Ch. 4!

2. Vectors: Some Topics in Ch. 1, Section 7
General Discussion.
Vector  A quantity with magnitude & direction.
Scalar  A quantity with magnitude only.
Here, we’ll mainly deal with Displacement & Velocity. But, our discussion is valid for any vector!
The Ch. 1 vector review has a lot of math!
It requires a detailed knowledge of trigonometry!
Problem Solving
A diagram or sketch is helpful & vital!
I don’t see how it is possible to solve a vector problem without a diagram!

3. Rectangular (Cartesian) A “Standard Set” of xy Coordinate Axes
Coordinate Systems
Rectangular (Cartesian)
Coordinates
“Standard” coordinate axes.
A point in the plane is (x,y)
If convenient, we could reverse + & -
-,+
+,+
-, -
+, -
A “Standard Set” of xy Coordinate Axes

4. Vector & Scalar Quantities
Vector  Quantity with magnitude & direction.
Scalar  Quantity with magnitude only.
Equality of Two Vectors
Consider 2 vectors, A & B
A = B means A & B have
the same magnitude & direction.

5. Vector Addition, Graphical Method
Addition of Scalars: Use “Normal” arithmetic!
Addition of Vectors: Not so simple!
Vectors in same direction: Use simple arithmetic

6. Vector Addition, Graphical Method
Addition of Scalars: Use “Normal” arithmetic!
Addition of Vectors: Not so simple!
Vectors in same direction: Use simple arithmetic
Example 1: Suppose we travel 8 km East on
day 1 & 6 km East on day 2.
Displacement = 8 km + 6 km = 14 km East

7. Vector Addition, Graphical Method
Addition of Scalars: Use “Normal” arithmetic!
Addition of Vectors: Not so simple!
Vectors in same direction: Use simple arithmetic
Example 1: Suppose we travel 8 km East on
day 1 & 6 km East on day 2.
Displacement = 8 km + 6 km = 14 km East
Example 2: Suppose we travel 8 km East on
day 1 & 6 km West on day 2.
Displacement = 8 km - 6 km = 2 km East
“Resultant” = Displacement

8. Adding Vectors in the
Same Direction:

9. Graphical Method of Vector Addition
For 2 vectors NOT along the gsame line, adding is more complicated:
Example: D1 = 10 km East
D2 = 5 km North.
What is the resultant (final) displacement?
2 Methods of Vector Addition:
Graphical (2 methods of this also!)
Analytical (TRIGONOMETRY)

10. Graphical Method of Adding Vectors
“Recipe”
Draw the 1st vector.

11. Graphical Method of Adding Vectors
“Recipe”
Draw the 1st vector.
Draw the 2nd vector
starting at the tip of the
first vector

12. Graphical Method of Adding Vectors
“Recipe”
Draw the 1st vector.
Draw the 2nd vector
starting at the tip of the
first vector
Continue to draw vectors “tip-to-tail”
The sum is drawn from the tail of the first vector to the tip of the last vector
Example:

13. Example: 2 vectors NOT along the same line. Figure!
D1 = 10 km E, D2 = 5 km N.
Resultant = DR = D1 + D2 = ?

14. Example: 2 vectors NOT along the same line. Figure!
D1 = 10 km E, D2 = 5 km N.
Resultant = DR = D1 + D2 = ?
In this special case ONLY, D1 is perpendicular to D2.
So, we can use the Pythagorean Theorem.
DR = 11.2 km
Note!
DR < D1 + D2
(scalar addition)

15. D1 = 10 km E, D2 = 5 km N. Resultant = DR = D1 + D2 = ? DR = 11.2 km
Note!
DR < D1 + D2
Graphical Method of Addition
Plot the vectors to scale, as in the figure.
Then measure DR & θ.
Results in DR = 11.2 km, θ = 27º N of E

16. Graphical Addition Recipe
This example illustrates general rules of graphical addition, which is also called the
“Tail to Tip” Method.
Consider R = A + B (See figure!).
Graphical Addition Recipe
1. Draw A & B to scale.
2. Place the tail of B at the
tip of A
3. Draw an arrow from the
tail of A to the tip of B
4. This arrow is the Resultant R
Measure its length & angle with the x-axis.

17. Order Isn’t Important! DR = D1 + D2 = D2 + D1
Adding vectors in the opposite order gives the same
result: In the example in the figure,
DR = D1 + D2 = D2 + D1
Figure 3-4. Caption: If the vectors are added in reverse order, the resultant is the same. (Compare to Fig. 3–3.)

18. Graphical Method of Vector Addition
Adding (3 or more) vectors:
V = V1 + V2 + V3
Even if the vectors are not at right angles, they can be added graphically with the tail-to-tip method.

19. Parallelogram Method V = V1 + V2
A 2nd Graphical Method of Adding Vectors
(equivalent to the tail-to-tip method, of course!)
V = V1 + V2
1. Draw V1 & V2 to scale from a common origin.
2. Construct a parallelogram using V1 & V2 as
2 of the 4 sides.
3. Resultant V = Diagonal of the
Parallelogram from a Common Origin
(measure length & the angle it makes with the x axis)
See Figure Next Page!

20. Parallelogram Method Mathematically, we can move vectors around
A common error!
Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect.
Mathematically, we can move vectors around
(preserving their magnitudes & directions)

21. Subtraction of Vectors
First, Define The Negative of a Vector:
- V  vector with the same magnitude (size) as V but with opposite direction.
Math: V + (- V)  0
Then add the negative vector.
For 2 vectors, V1 & V2:

22. Subtracting Vectors
To subtract one vector from another, add the first vector to the negative of the 2nd vector, as in the figure below:

23. Multiplication by a Scalar
A vector V can be multiplied by a scalar c
V' = cV
V'  vector with magnitude cV & same direction as V.
If c is negative, the resultant is in the opposite direction.

24. Example Answers: Length = 48.2 km β = 38.9º
Consider a 2 part car trip:
Displacement A = 20 km due North.
Displacement B = 35 km 60º West of North.
Find (graphically) resultant displacement vector R
(magnitude & direction). R = A + B. See figure below.
Use a ruler & protractor to find the length of R
& the angle β.
Answers:
Length = 48.2 km
β = 38.9º