1. Chapter 3: Vectors

2. Two Dimensional Vectors Algebraic Vector Operations
Outline
Two Dimensional Vectors
Magnitude & Direction
Algebraic Vector Operations
Equality of vectors
Vector addition
Multiplication of vectors with scalars
Scalar Product of Two Vectors
(a later chapter!)
Vector Product of Two Vectors

3. Vectors: General discussion
Vector  A quantity with magnitude & direction.
Scalar  A quantity with magnitude only.
Here, we mainly deal with
Displacement  D & Velocity  v
Our discussion is valid for any vector!
This chapter is mostly 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!

4. Coordinate Axes A point in the plane is denoted as (x,y)
Usually, we define a reference frame using a standard coordinate axes. (But the choice of reference frame is arbitrary & up to us!). Rectangular or Cartesian Coordinates:
2 Dimensional “Standard” coordinate axes.
A point in the plane is denoted as (x,y)
Note, if its convenient, we could reverse + & - !
Standard sets of xy (Cartesian or rectangular)
coordinate axes
- ,+
+,+
- , -
+ , -

5. Plane Polar Coordinates Trigonometry is needed to understand these!
A point in the plane is denoted as (r,θ) (r = distance from origin, θ = angle from the x-axis to a line from the origin to the point).
(a) (b)

6. Equality of two vectors
2 vectors, A & B.
A = B means that A & B
have the same magnitude & direction.

7. Vector Addition, Graphical Method
Addition of scalars: “Normal” arithmetic!
Addition of vectors: Not so simple!
Vectors in the same direction:
Can also use simple arithmetic
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

8. Adding Vectors in the Same Direction

9. 2 methods of vector addition:
Graphical Method
For 2 vectors NOT along the same 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. For 2 vectors NOT along same line:
D1 = 10 km E, D2 = 5 km N.
We want to find the Resultant = DR = D1 + D2 = ?
In this special case ONLY, D1 is perpendicular to D2.
So, we can use the Pythagorean Theorem.
Note! DR < D1 + D2
(scalar addition)
DR = 11.2 km
The Graphical Method requires measuring
the length of DR & the angle θ. Do that & find
DR = 11.2 km, θ = 27º N of E

11. That arrow is the Resultant R
This example illustrates the general rules (for the “tail-to-tip” method of graphical addition).
Consider R = A + B:
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
That arrow is the Resultant R
(measure the length & the angle it makes with the x-axis)

12. Order isn’t important!
Adding vectors in the opposite order gives the same result. In the example, 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.)

13. Graphical Method Continued
Adding 3 (or more) vectors
V = V1 + V2 + V3
Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

14. Graphical Method V = V1 + V2
A 2nd graphical method of adding vectors:
(100% equivalent to the tail-to-tip method!)
V = V1 + V2
1. Draw V1 & V2 to scale from a common origin.
2. Construct a parallelogram with V1 & V2 as 2 of the 4 sides.
Then, the
Resultant V = The diagonal of the parallelogram from the common origin
(measure the length and the angle it makes with the x-axis)

15. Mathematically, we can move vectors around
So, The Parallelogram Method may also be used for the graphical addition of vectors.
Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect.
A common error!
Mathematically, we can move vectors around
(preserving magnitudes & directions)

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

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

18. Example 3.2 Use a ruler & protractor to find the length of R
A two part car trip:
First displacement: A = 20 km due North.
Second displacement B = 35 km 60º West of North.
Find (graphically) the resultant displacement vector R (magnitude & direction): R = A + B
Use a ruler & protractor to find the length of R
& the angle β:
Length = 48.2 km
β = 38.9º