3.2 Vector Addition and Subtraction: Graphical Methods

We will see how to resolve the vectors in the following methods.

Many areas of physics will benefit from such techniques.

You can learn about position, velocity and acceleration.

The ladybug can be moved by setting the position, speed or acceleration.
To analyze the behavior, choose linear, circular or elliptical motion.

This one of the Hawaiian Islands has a scale map that can be used to determine displacement.

A journey from Hawai'i to Moloka'i has a number of legs.
The total twodimensional displacement of the journey can be determined graphically with a ruler.
For example, displacement, velocity, acceleration, and force are all vectors.
In one-dimensional, straight-line, motion, a plus or minus sign can be used.
In two dimensions (2-d), we use an arrow to point in the direction of the vector relative to the reference frame.

An example of a graphical representation of a vector is shown in Figure 3.9, which shows the total displacement for a person in a city.

A boldface symbol, such as, stands for a vector.
The symbol in italics represents its magnitude and direction.

A boldface variable will be represented in this text.

The quantity force will be represented with the vector, which has both magnitude and direction.
The direction of the variable will be given by an angle and the magnitude will be represented by a variable in italics.

A person is walking 9 blocks east and 5 blocks north.

The displacement is north of the east.

Draw a line at an angle to the east-west axis.

The ruler is used to measure the length of the arrow.
The direction is north of east, and the magnitude is 10.3 units.

The head of the first, east-pointing vector should be the location of the tail.

The length of the arrow is determined by the magnitude of the vector.
The angle with respect to the east is measured with a protractor.

Use a ruler and protractor to draw an arrow to represent the first block.

Draw an arrow to represent the second vector.

Continue this process if there are more than two.

In our example, we have only two, so we have finished placing the arrows tip to tail.

Only the precision of the measuring tools and the accuracy of the drawings can limit the graphical addition of vectors.

It's valid for all of them.

You can use a graphical technique to find the total displacement of a person who walks on a flat field.

She walks 25.0 m north of east.
She walks north of east.
She turned and walked 32.0 m south of east.

The lengths of the displacements are proportional to the distance and directions specified relative to the east-west line.

The head-to-tail method will give a way to determine the magnitude and direction of the displacement.

The easiest way to measure the angle between the vector and the nearest horizontal or vertical axis is to use a compass.

The protractor is flipped upside down to measure the angle between the east and west pointing axes.

The total displacement is seen to have a magnitude of 50.0 m and lie south of east.

It is important to note that the resulting is not related to the order in which the vectors are added.

The result is the same when the same vectors are added in a different order.

In every case, this characteristic is true.
In any order, they can be added.

A simple extension of addition is called Vector subtraction.

We must first define what we mean by subtraction.
It is the same length as but points in a different direction.
We flip the vector so it points in a different direction.

The negative of a vector is the same as the positive and points in the opposite direction.

The negative has the same length but in a different direction.

The addition of to is simply defined as the subtraction of vector.

The addition of a negative vector is called vector subtraction.
The results are unaffected by the order of subtraction.

This is similar to the removal of scalars.

The result is not dependent on the order in which the subtraction is made.
The following example illustrates how the techniques outlined above are used.

A woman is sailing a boat at night.

The instructions say to first sail 27.5 m in a direction north of east from her current location and then travel 30.0 m in a direction north of east or west of north.
The location of the dock is compared with this location.

The first and second legs of the trip can be represented with a vector.

There is a dock at the location.
If the woman travels in the opposite direction for the second leg of the journey, she will travel a distance south of east.
The vector is in the opposite direction.

The location of the dock will be compared with the location at which the woman mistakenly arrives.

If the woman travels in the opposite direction for the second leg of the trip, she will end up a long way from the dock.

The graphical method of subtracting a vectors works the same as the addition method.

If we decided to walk three times as far on the first leg of the trip, we would walk in a direction north of east.

The direction stays the same as the magnitude changes.

If the scalar is negative, you can change the vector's magnitude and give a new one the opposite direction.

If you add -2 to the equation, the magnitude doubles but the direction changes.
The following rules are summarized in the following way: If the direction of the vector is positive, the magnitude of it becomes the absolute value, and if it is negative, the direction is reversed.

In a lot of situations, the numbers are multiplied by the numbers.

The inverse of multiplication is division.
The difference is that dividing by 2 is the same as dividing by the value.
The rules for division are the same as for multiplication of vectors by scalars.

In the examples above, we have been adding more than one factor to the equation.

We will need to do the opposite in many cases.
We need to find what other vectors add together to produce it.

If we know the total displacement of a person walking in a city, we can figure out how many blocks north and east they had to walk.

The inverse of the process followed to find the total displacement is the method of finding the components of the displacement in the east and north directions.
It is an example of how to find components of a picture.
This is a useful thing to do in many physics applications.
Right triangles are involved in most of these because they involve finding components along parallel axes.
The analytical techniques presented in the book are ideal for finding components.

In the "Arena of Pain", you can learn about position, velocity, and acceleration.

The green arrow can be used to move the ball.
The game will be more difficult if more walls are added to the arena.
Make a goal as fast as you can.