Chapter 3: Vectors
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
(a later chapter!)
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!
Coordinate Axes
• 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
- ,+
-,-
+,+
+,-
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)
Equality of two vectors
2 vectors, A & B.
A = B means that A & B
have the same magnitude &
direction.
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
Adding Vectors in the Same Direction
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)
• 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
• 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)
Order isn’t important!
Adding vectors in the opposite order gives the same
result. In the example, DR = D1 + D2 = D2 + D1
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.
Graphical Method
• 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)
So, The Parallelogram Method may also be used
for the graphical addition of vectors.
A common error!
Mathematically, we can move vectors around
(preserving magnitudes & directions)
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)
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.
Example 3.2
• 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