Chapter 3: Vectors
OutlineTwo Dimensional Vectors
Magnitude & DirectionAlgebraic Vector Operations
Equality of vectorsVector addition
Multiplication of vectors with scalarsScalar Product of Two Vectors
(a later chapter!)Vector Product of Two Vectors
(a later chapter!)
Vectors: General discussionVector 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 CoordinatesTrigonometry 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 vectors2 vectors, A & B.
A = B means that A & Bhave 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 arithmeticExample: 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.
The Graphical Method requires measuring the length of DR & the angle θ. Do that & find
DR = 11.2 km, θ = 27º N of E
Note! DR < D1 + D2
(scalar addition)DR = 11.2 km
• 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 A3. 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.
• 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)
Graphical Method
So, The Parallelogram Method may also be used for the graphical addition of vectors.
Mathematically, we can move vectors around (preserving magnitudes & directions)
A common error!
Subtraction of Vectors • First, Define the Negative of a Vector:
-V the vector with the same magnitude (size) as Vbut 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