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Vectors

Purpose

1. To study vectors and their addition graphically.

2. To study resolution of a two dimensional vector and the addition method through components.

Introduction

Vectors are quantities that have both direction and magnitude. The magnitude is the numerical value of the vector. A

quantity that has only magnitude (no direction) is called a scalar. A scalar could have any value including negative values,

but it does not have a direction. Examples of scalars are temperature, distance and mass. Examples of vectors are force,

velocity and acceleration. The direction of a force vector, is the direction in which the force is applied. The magnitude of

the force vector is the strength of the force. We draw vectors as ray with the direction as the arrow. When drawing a

vector we represent the magnitude of a vector by the length of that vector. So a strong force is represented by a longer

vector than a weak force.

As a notation, we will write a vector in bold ‘𝑨’, while its magnitude in normal font ‘𝐴’.

Part 1: Adding vectors

When we add vectors we have to take the direction of the vector into consideration. Two equal but opposite vectors

add up to zero. Since a vector is characterized by only magnitude (represented by length of the vector) and direction,

then we if we move the vector parallel to itself while keeping its length fixed, we still have the same vector (since in

moving the vector parallel to itself we preserved its direction and magnitude). A method to add vectors is to arrange

them in a way that the tip of the 1st vector is at the tail of the 2nd vector and the tip of the 2nd vector is at the tail of the

3rd vector and so on. Then the sum of the vectors, called Resultant vector, 𝑹, is a vector that connects the tail of the 1st

vector to the tip of the last vector. See figure 1. 𝑹 = 𝑨1 + 𝑨2 + 𝑨3. If a

vector 𝑬 is equal and opposite to 𝑹 (𝑬 = −𝑹), then 𝑬 + 𝑹 = 0. 𝑬 is called

the ‘equilibrant vector’. If the vectors represent forces, then 𝑬 is the force

that when added to the other force vectors will produce equilibrium.

Part 2: Resolving a vector

Consider the two vectors 𝑨𝑥 (horizontal) and 𝑨𝑦 (vertical) in Figure 2. If we add the vectors 𝑨𝑥 and 𝑨𝑦 using the

graphical addition method above, we get the Resultant vector. We will call it 𝑨 here. So, 𝑨 = 𝑨𝑥 + 𝑨𝑦. So in a plane,

any vector 𝑨 can be analyzed (resolved) to two components a horizontal component 𝑨𝑥 and a vertical component 𝑨𝑦.

Using trigonometry,

𝐴𝑥 = 𝐴𝑐𝑜𝑠𝜃 and 𝐴𝑦 = 𝐴𝑠𝑖𝑛𝜃 𝑒𝑞𝑛𝑠. (1)

𝑡𝑎𝑛𝜃 =𝐴𝑦

𝐴𝑥 𝑒𝑞𝑛. (2)

Using Pythagoras theorem 𝐴 = 𝐴𝑥2 + 𝐴𝑦2 𝑒𝑞𝑛. (3)

In order to add vectors, there is a method other than the graphical method mentioned above. This other method uses

vector resolution. First we resolve all the vectors to their x and y components then we add the x components (with their

signs) and we add the y components (with their signs) then find the magnitude of the resultant vector using eqn. 3, and

its direction using eqn. 2.

Figure 1: Adding vectors graphically

𝑨1 𝑨2

2 𝑨3

𝑨1

𝑨2

𝑨3

𝑹

Figure2: Resolving a vector 𝑨

𝑨𝑥

𝑨𝑦 𝑨𝑦

𝑨𝑥

θ 𝑨

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Running the experiment The data sheet is on page 5 & the graph paper is on page 6

Part 1: Adding vectors graphically and finding the equilibrium vector (Ask the instructor for your set of 𝑨𝟏, 𝑨𝟐, & 𝐴𝟑)

If you do not have a protractor you may find these links useful (copy the link and paste it in your browser):

i) https://www.ginifab.com/feeds/angle_measurement/

ii) https://www.inchcalculator.com/wp-content/uploads/2016/04/protractor.pdf

1) Consider figure 3. Each square has a side equal 1 cm. Using the graphical method for adding vectors find the resultant

𝑹 of the 3 vectors 𝑨1, 𝑨2, and 𝑨3. The tails of the vectors are at point (8, 8), & for their tips: ask your instructor. In this

graphical method you can, using a ruler, a protractor (or using the coordinates: for ex. 𝑨2 has a tip that is 5 units left & 5

units up from from its tail) & the graph paper given on page 6 here (follow the orientation given in the graph paper: 𝑥-

horizontal & 𝑦-vertical), to redraw 𝑨2 so that its tail is at the tip of 𝑨1, and 𝑨3 so that its tail is at the tip of 𝑨2, then

connect the tail of 𝑨1 to the tip of 𝑨3: this is the resultant vector, 𝑹. See figures 4 and 5 for a method of redrawing the

vectors using the protractor with same angles (In fig. 4. the magnitudes are not to scale). With the ruler you draw the

vector with the same length (this represent same magnitude) and with the protractor you draw the vectors with the

same directions as demonstrated in figure 5. 𝑭1, 𝑭2, 𝑎𝑛𝑑 𝑭3 are examples to resemble 𝑨1, 𝑨2 𝑎𝑛𝑑 𝑨3, respectively.

2) Using the pencil, ruler and the protractor, find the magnitude and direction of 𝑹. Then find 𝑬 = −𝑹.

Figure 3: Adding vectors graphically

𝑨1 𝑨2

𝑨3

β

γ

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3) Open the simulator https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_en.html

Click Lab. There are 2 sets of vectors, blue and orange. We are going to use only the blue set of vectors. Drag a blue

vector (we will call it 𝑨1) to the graph sheet of the simulator (see fig. 3 above), so that its tail is at coordinates (x, y) = (8, 8)

& its tip is at point (16, 14). Click values. The magnitude of the vector should show as 10 (default). Drag another blue

vector (we will call it 𝑨2) to the graph sheet, place it such that its tail is at the same tail of vector 𝑨1; tail at point (8, 8),

and tip of vector 𝑨2 is at (3, 13). The magnitude of 𝑨2 should be 7.1. Drag a third blue vector (we will call it 𝑨3) to the

graph sheet. Place it such that its tail is coinciding with the tail of 𝑨1 and 𝑨2 at (8, 8), and its tip is at (4, 5). Its magnitude

should be 5. Assume each square of the graph has side length equal 1centimeter, 1 cm.

4) Click Sum (for the blue vectors) at the top right of the simulator. This shows the sum vector 𝑺 for the blue vectors,

which up till now are only the three blue vectors 𝑨𝟏, 𝑨𝟐 and 𝑨𝟑, we will call the resultant 𝑹. Drag this Sum vector, 𝑺 so

that its tail is at (8, 8). Click angle . This shows the direction. If it is difficult to see the values, you can uncheck the

values option at the top right of the simulator and move the mouse to the vector that you want to select & click the

vector, the magnitude of the vector & all its values will show above the graph. You can also alternatively keep the values

and angles options checked & drag the sum vector a bit away to see the values clearly.

Compare to your calculated 𝑹 in step 2 above. Compute the calculated 𝑬 = −𝑹, (Notice that magnitudes of 𝑬 & 𝑹 are

the same, but the direction of 𝑬 is opposite to the direction of 𝑹. So the angle of 𝑬, is 𝜃𝐸 counter clock wise, (CCW) from

the positive x-axis is angle 𝜃𝑅 of 𝑹 plus 180o, 𝜃𝑅 + 180𝑜 and from below the positive x-axis (clock wise, CW) the angle of

𝑬 is −𝜃𝐸′ = −[360𝑜 − (𝜃𝑅 + 180𝑜)] ). Record these values for 𝑹 and 𝑬 in the table of part 1, step 4) in the data sheet,

we will use them for the next step 5. In this table of part 1, record both 𝜃𝐸(CCW) and −𝜃𝐸′ (CW), from the positive x-axis.

5) Drag a fourth blue vector to represent the vector 𝑬 that we calculated in step 4, and place it with its tail at (8, 8),

magnitude as calculated in step 4 (= magnitude of 𝑹 found and recorded in step 4) and angle clock wise from the

positive x-axis equal to −𝜃𝐸′ = −[360𝑜 − (𝜃𝑅 + 180𝑜)] such that it is opposite to 𝑹 that was found in step 4. Note that

as you place the fourth vector 𝑬 and while you are adjusting its magnitude and direction (angle), the sum vector, 𝑺

changes. Can you explain why? (Hint: what does the sum vector, 𝑺 represent?). After you adjust the magnitude and

direction (angle) of the fourth vector 𝑬, what is the value of the new sum vector, 𝑺 ? Explain.

Part 2: Resolving a vector

Figure 6: Resolving a vector

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1) For the vector shown in Figure 6, (we will call it vector 𝒂), using equations 1 in the introduction, find the x-component

𝒂𝑥 and the y-component 𝒂𝑦.

2) In the same simulator of part 1 (do not remove the vectors we will use them again in part 3), click ‘Explore 2D’ at the

bottom of the window. Drag a blue vector 𝒂 and place it such that its tail is at the origin (as shown in figure 6). Click

values and click angle. Adjust the vector 𝒂 so that its magnitude is 10.4 and its angle counter clock wise from the positive

x- axis is 106.7o, as shown in figure 6.

3) In components click the icon so as to show the x and y components. Record the values and compare with

your calculated values in step 1.

Part 3: Adding vectors using components

1) In the same simulator, click ‘Lab’ at the bottom of the window. You should see your vectors 𝑨1, 𝑨2 , 𝑎𝑛𝑑 𝑨3 from part

1. You can remove the 4th vector 𝑬 (drag it back to the blue vector box at the right). Using the magnitude and angle of

each vector, calculate the x and y components: 𝐴1𝑥 , 𝐴1𝑦 , 𝐴2𝑥 , 𝐴2𝑦 , 𝐴3𝑥 , 𝐴3𝑦 , and 𝑅𝑥& 𝑅𝑦 (you may need to uncheck

the values option, in order to see clearly the angles). Note that in equations 1, θ is the angle measured counter clock

wise from the positive x-axis. So, for example, for the vector 𝑨3, in fig. 7, the angle 𝜃𝐴3 is 360𝑜 – 143.1𝑜 .

2) Click components (you can uncheck the angles option and select the values option to see the values of the

components clearly. You can also drag the vectors a bit apart to see the components of each vector clearly. See fig. 7).

Record the values of 𝐴1𝑥 , 𝐴1𝑦 , 𝐴2𝑥 , 𝐴2𝑦 , 𝐴3𝑥 , 𝐴3𝑦 , and 𝑅𝑥& 𝑅𝑦 and compare to your calculated values in step 1.

3) a) Add the x components (including sign) and b) add the y components (including sign). Record your calculated x and y

components.

4) Compare your calculated values in step 3 to the values displayed by the simulator for 𝑅𝑥& 𝑅𝑦 .

Question:

1) State the condition for equilibrium in two ways (i) as illustrated in part 1, (ii) as illustrated in part 3.

Figure 7: Adding vectors using components

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Data sheet

Name: Group: Date experiment performed:

Part 1: Adding vectors graphically

Step 2) magnitude of 𝑹: Direction of 𝑹, 𝜃𝑅: magnitude of 𝑬: Direction of 𝑬, 𝜃𝐸:

Step 4)

Measured 𝑹 Calculated 𝑬 = −𝑹 (use measured 𝑹 of step 4)

magnitude Direction 𝜃𝑅 CCW from +ve x-axis

magnitude Direction 𝜃𝐸 CCW from +ve x-axis

Direction 𝜃𝐸′ CW from

+ve x-axis

Step 5) Why does the sum vector, 𝑺 change as you add and while adjusting the fourth vector, 𝑬?

With the fourth vector, 𝑬 added, the sum vector 𝑺 = Explanation:

Part 2: Resolving a vector

Step 1) 𝑎𝑥 = direction of 𝑎𝑥 : 𝑎𝑦 = direction of 𝒂𝑦 :

Step 3)

Measured 𝒂𝑥 Measured 𝒂𝑦

magnitude direction magnitude direction

Part 3: Adding vectors using components

Step 1) Calculated:

𝐴1𝑥 𝐴1𝑦 𝐴2𝑥 𝐴2𝑦 𝐴3𝑥 𝐴3𝑦 𝑅𝑥 𝑅𝑦

Step 2) Measured:

𝐴1𝑥 𝐴1𝑦 𝐴2𝑥 𝐴2𝑦 𝐴3𝑥 𝐴3𝑦 𝑅𝑥 𝑅𝑦

Step 3) a) Addition of x components, ∑𝐴𝑥 :

b) Addition of y components, ∑𝐴𝑦 :

Step 4) a) Compare ∑𝐴𝑥 to measured 𝑅𝑥 :

b) Compare ∑𝐴𝑦 to measured 𝑅𝑦 :

Answer to question 1:

(The graph paper is on the next page, page 6. Each square is 1cm x 1cm).

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