Vector Addition and Laws of Vector Composition

Study Notes

Vector Addition

Vectors represent physical quantities that have both magnitude and direction
The sum of two or more vectors is called the resultant vector
Vectors must have the same units to be added correctly
There are two common methods for finding the vector sum: the graphical method and the analytical (algebraic) method

Triangle Law

When two or more vectors are added, the units must be the same
The final result is the same as if the particle had started the same initial point and undergone a single displacement
The vector sum is represented by $ \vec{S} = \vec{S_1} + \vec{S_2}$
Vectors are placed head to tail
The arrow from the tail of the first vector to the head of the second vector represents the resultant

Parallelogram Law (and Polygon Law)

The vectors are placed at the same initial point, forming the sides of a parallelogram
The diagonal of the parallelogram represents the resultant vector
The Polygon Law extends this to adding multiple vectors

Activity 2.1

Categorize physical quantities as scalars or vectors
Area, Weight, Pressure, and Density are Scalar quantities.
Distance is a Vector quantity.

Representation of Vectors

Vectors can be represented graphically by an arrow
The length of the arrow represents the magnitude of the vector
The tip of the arrow represents the direction of the vector
In diagrams, a scale similar to maps is used

Activity 2.2

Ask a friend to walk a certain distance on a volleyball court.Measure the distance and starting point.Use the starting point as origin of coordinates (x-axis = East, y-axis = North).Graphically show the displacement on graph paper.

Types of vectors

Vectors in the same direction are parallel vectors.
Antiparallel vectors have opposite directions, whether their magnitudes are the same or not.
If two vectors represent the same quantity, and have the same magnitude and direction regardless of their location, they are equal vectors.

Collinear and Coplanar Vectors

Collinear vectors: Vectors lying on the same line or lines parallel to each other.
Coplanar vectors: Three or more vectors lying on the same plane or planes parallel to each other.

Zero Vector

A zero vector is a vector with a magnitude of zero.
Its starting point coincides with its terminal point.

Orthogonal Vectors

Orthogonal vectors are perpendicular to one another.

Unit Vector

A unit vector is a vector with a magnitude of 1 (unit length).

Negative of a Vector

The negative of a vector has the same magnitude as the original vector, but its direction is opposite.
If vector $\vec{A}$ is the original vector, the negative of $\vec{A}$ is denoted as $-\vec{A}$.
If $\vec{S} = 60$ km (North) then $-\vec{S} = 60$ km (South)
If $\vec{S} = 100$km/h (200°South of East), then $-\vec{S} = 100$ km/h (200° North of West)

Position Vector

A position vector symbolizes the position or location of a point.
It can represent a point in space or locate a point by specifying its position $\vec{OA}$ from the origin O to a point.

Activity 2.3

This activity involves a parallelogram law experiment.
This activity might involve graphing vectors on a coordinate plane.
This activity might require calculations.

Activity 2.4

This activity encourages discussion with classmates.
Discussing the error of assuming the magnitude of C (C = A + B) is equal to the sum of the magnitudes of A and B.
Discussing the Polygon Law of Vector Addition.

Activity 2.5

This experiment involves adding individual displacement vectors.
The experiment has 8 steps, from marking starting points to creating a vector diagram on graph paper.

Forces in Hydro Power Dams

Figure 2.2 illustrates a diagram of a hydro power dam.
The labeled forces acting on the dam are head water, weight of dam, and uplift.

Force Vectors in Action

Figure 2.3a shows a person rowing a canoe.
Figure 2.3b shows a tug-of-war.
Vectors play an important role in analyzing forces.

Vector Addition Methods

There are two common methods for finding the vector sum of two or more vectors:
- Graphical Method
- Analytic (algebraic) Method

Resultant Vector

The resultant vector is the vector sum of two or more vectors.

Geometric Method

The geometric method for adding and subtracting vectors is explained in the text.

Graphical Method of Vector Addition (2-D)

The graphical method is described in the document.
You should be able to define the term resultant vector, explain the geometric method and apply the geometric method for addition of two vectors in two dimensions.

Example

Find the negative of each of the following vectors

a.displacement vector = 60 km toward North = 60 km toward South
b.Velocity vector = 100 km/h towards 200° south of east = 100 km/h towards 200° north of west.

Figure 2.1

Figure 2.1 displays a vector represented by an arrow drawn to scale.
The tools used are a ruler, protractor, and graph paper.
Note: the figure itself is not directly mentioned in the text.

Description

Explore the principles of vector addition, including both the Triangle and Parallelogram Laws. This quiz covers how vectors can be combined and the importance of maintaining consistent units. Test your understanding of graphical and analytical methods in finding resultant vectors.

Vector Addition and Laws of Vector Composition

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Questions and Answers

What is the correct scale to use for creating a vector diagram?

Subtracting a vector is the same as adding a vector of the same magnitude but in the same direction.

What is the minimum number of vectors required to produce a zero resultant?

The vector __________ has the same magnitude as B but the opposite direction.

Match the following terms with their correct descriptions:

What does the diagonal of a parallelogram represent when adding two vectors graphically?

The magnitude of the resultant vector C is always equal to the sum of the magnitudes of vectors A and B when C = A + B.

What is the result of adding a vector to the negative of itself?

In the polygon law of vector addition, each vector is placed with its _____ at the head of the previous vector.

Match the following terms related to vector addition with their definitions: