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