Addition of Vectors By Means of Components - Physics

Summary

This video script offers an insightful tutorial on vector addition, emphasizing the importance of both magnitude and direction. It explains the straightforward process of adding parallel vectors and the more complex task of combining vectors at various angles, including perpendicular ones. The script delves into calculating the resultant force vector's magnitude and direction through component analysis and trigonometric methods, providing clear examples and formulas to guide viewers through the process.

Takeaways

• 📚 A vector is a quantity with both magnitude and direction, such as a force vector with a specific strength and direction.
• 🔍 When adding parallel vectors, simply sum their magnitudes to get the resultant force.
• 🤔 For anti-parallel vectors, subtract the smaller magnitude from the larger to find the net force, with the direction of the larger vector.
• 📐 To find the resultant of perpendicular vectors, calculate the square root of the sum of the squares of their magnitudes.
• 🧭 The direction of the resultant vector can be found using the inverse tangent function (arctan) of the y-component divided by the x-component.
• 📈 The magnitude of the resultant vector from non-parallel or non-perpendicular vectors is the square root of the sum of the squares of the individual x and y components.
• 📉 To determine the direction of the resultant vector, use the arctan of the y-component over the x-component, adjusting for the correct quadrant.
• 📝 The reference angle can be found by taking the arctan of the y-component divided by the x-component, always resulting in an acute angle.
• 📈 The component method involves breaking down vectors into their x and y components, summing these components, and then finding the resultant vector.
• 📐 The Pythagorean theorem is used to relate the magnitude of a vector to its x and y components.
• 🔢 The angle theta of a vector can be determined using trigonometric relationships, such as arctan for the direction.

Q & A

What is a vector and what are its two main components?

A vector is a quantity that has both magnitude and direction. The magnitude is the size of the vector, such as the strength of a force, while the direction indicates the orientation, like east or north.

How do you add two vectors that are parallel to each other?

When adding two parallel vectors, you simply add their magnitudes together. For example, if you have a 100 Newton force vector directed east and another 50 Newton force vector in the same direction, the resultant force is 150 Newtons.

What is the resultant vector when adding two anti-parallel vectors?

For anti-parallel vectors, you subtract their magnitudes. For instance, a 200 Newton vector directed east and a 120 Newton vector directed west would result in a net force of 80 Newtons directed east.

How do you determine the magnitude and direction of the resultant vector when adding two perpendicular vectors?

For perpendicular vectors, you find the magnitude of the resultant vector by calculating the square root of the sum of the squares of their magnitudes. The direction can be found using the inverse tangent of the y-component divided by the x-component.

What is the resultant force vector and its direction when adding a 60 Newton east vector and a 90 Newton west vector?

The resultant force vector would be 30 Newtons directed west, as the east vector is 60 Newtons and the west vector is 90 Newtons, resulting in a net force of -30 Newtons, which can be stated as 30 Newtons west.

How do you calculate the magnitude of the resultant vector when adding a 50 Newton west vector and a 120 Newton south vector?

You would calculate the magnitude by taking the square root of the sum of the squares of the two vectors, which in this case is the square root of 50 squared plus 120 squared, resulting in a magnitude of 130 Newtons.

What is the reference angle and how is it used to determine the direction of a resultant vector?

The reference angle is the acute angle formed between the resultant vector and the x-axis, found using the arctangent of the y-component divided by the x-component. It is used to adjust the direction of the resultant vector based on the quadrant it lies in.

How do you find the direction of the resultant vector when adding a 50 Newton west vector and a 120 Newton south vector?

First, find the reference angle using the arctangent of the y-component (120) divided by the x-component (50), which gives an angle of 67.4 degrees. Since this is in quadrant three, the direction of the resultant vector is 180 degrees plus the reference angle, resulting in 247.4 degrees from the positive x-axis.

What is the magnitude and direction of the resultant force vector when adding a 45 Newton east vector and a 60 Newton south vector?

The magnitude is found by taking the square root of the sum of the squares of the two vectors, which is 75 Newtons. The direction is found by calculating the angle in the fourth quadrant, which is 360 degrees minus the reference angle of 53.1 degrees, resulting in 306.9 degrees from the positive x-axis.

How do you add vectors that are not parallel or perpendicular using the component method?

You break down each vector into its x and y components based on its magnitude and angle. Then, you add the x-components together and the y-components together. Finally, you find the magnitude of the resultant vector using the Pythagorean theorem and the direction using the inverse tangent of the y-component divided by the x-component.

Can you provide an example of adding a 100 Newton east vector and a 150 Newton vector at 30 degrees above the x-axis using the component method?

First, find the x and y components of each vector. For the 100 Newton east vector, the x-component is 100 and the y-component is 0. For the 150 Newton vector at 30 degrees, the x-component is 150 times cosine(30) and the y-component is 150 times sine(30). Add the x-components to get 229.9 and the y-components to get 75. Then, find the magnitude of the resultant vector, which is the square root of 229.9 squared plus 75 squared, resulting in 241.8 Newtons. The direction is found using the inverse tangent of the y-component divided by the x-component, giving an angle of 18.1 degrees.

Outlines

00:00

📚 Basic Vector Addition

This paragraph introduces the concept of vector addition, explaining that a vector has both magnitude and direction. It uses the example of force vectors to illustrate how to add vectors that are parallel, such as combining two forces in the same direction to get a net force. The paragraph also addresses the addition of anti-parallel vectors, where the resultant vector is the difference of their magnitudes, with the direction of the larger vector.

05:01

🔍 Resultant Vectors with Direction

The second paragraph delves into finding the resultant vector when combining vectors with different directions. It uses examples of east and west directed forces to demonstrate how to calculate the net force and its direction. The paragraph explains that the direction of the resultant vector is determined by the vector with the greater magnitude and introduces the concept of using the inverse tangent function to find the angle of the resultant vector relative to a given axis.

10:01

📐 Vector Addition in Quadrants

This paragraph discusses the process of adding vectors that are neither parallel nor anti-parallel, such as vectors directed in different quadrants. It explains how to find the magnitude of the resultant force vector by using the Pythagorean theorem and how to determine the direction of the resultant vector by calculating the reference angle and adjusting it based on the quadrant in which the resultant vector lies. The paragraph provides a step-by-step approach to solving these types of vector addition problems.

15:10

📘 Component Method for Vector Addition

The final paragraph introduces the component method for adding vectors that are not aligned with the coordinate axes. It explains how to break down a vector into its x and y components using trigonometric functions and how to sum these components to find the resultant vector. The paragraph also demonstrates how to find the magnitude and direction of the resultant vector by using the Pythagorean theorem and the inverse tangent function, respectively, and provides an example involving vectors with different magnitudes and directions.

Keywords

💡Vector

A vector is a mathematical quantity that has both magnitude (size) and direction. In the context of the video, vectors are used to represent forces with specific magnitudes and directional orientations. For example, a force vector of '100 newtons directed towards the east' demonstrates the concept of a vector with a magnitude of 100 and a direction to the east.

💡Magnitude

Magnitude refers to the size or length of a vector. In the video, when discussing the addition of vectors, the magnitude is the numerical value that represents the strength of the force. For instance, when adding two forces of 100 and 50 newtons, the magnitude of the resultant vector is 150 newtons.

💡Direction

Direction is the orientation of a vector in space. In the video, direction is crucial for understanding the resultant of vector addition. For example, two vectors with the same direction (both directed east) can be added by simply summing their magnitudes, resulting in a net force with the same direction.

💡Resultant Sum

The resultant sum is the outcome of adding two or more vectors together. It is the combined effect of the original vectors, taking into account both magnitude and direction. In the script, when two eastward forces of 100 and 50 newtons are added, the resultant sum is a net force of 150 newtons to the east.

💡Parallel Vectors

Parallel vectors are vectors that are aligned along the same or opposite directions. In the video, adding parallel vectors is straightforward, as their magnitudes can be directly added together. An example given is adding a 200 newton vector directed east with a 120 newton vector directed west, resulting in a net force of 80 newtons to the east.

💡Perpendicular Vectors

Perpendicular vectors are vectors that are at right angles to each other. The video explains that to find the resultant force vector of two perpendicular vectors, one must use the Pythagorean theorem to find the length of the hypotenuse of the right triangle formed by the vectors, as demonstrated with a 30 newton eastward vector and a 40 newton northward vector.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. In the context of the video, when adding perpendicular vectors, the magnitude of the resultant vector is equal to the length of the hypotenuse of the triangle formed by the original vectors, such as the square root of 30^2 + 40^2, which equals 50 newtons.

💡Component Method

The component method involves breaking down a vector into its horizontal (x) and vertical (y) components. In the video, this method is used to add vectors that are not parallel or perpendicular by finding the sum of the x and y components separately and then calculating the magnitude and direction of the resultant vector from these sums.

💡Arc Tangent

Arc tangent, often abbreviated as atan or tan^(-1), is a trigonometric function used to find the angle whose tangent is a given number. In the video, it is used to calculate the direction of the resultant vector by finding the angle between the x-axis and the vector, using the formula atan(y_component / x_component).

💡Quadrants

Quadrants refer to the four equal areas formed by the intersection of the x and y axes on a Cartesian plane. The video explains how to determine the direction of the resultant vector based on which quadrant it lies in, using the reference angle and adjusting it according to the specific quadrant rules.

Highlights

Introduction to vector addition, emphasizing magnitude and direction.

Explanation of adding parallel vectors by simply summing their magnitudes.

Demonstration of calculating the resultant force when adding vectors with opposite directions.

Illustration of the resultant vector's magnitude and direction when combining east and west directed forces.

Use of the Pythagorean theorem to find the resultant force of perpendicular vectors.

Application of trigonometry to determine the direction of the resultant force vector.

Example of calculating the magnitude and direction of a resultant force vector combining north and south directed forces.

Introduction of the component method for adding non-parallel or non-perpendicular vectors.

Formula for calculating the x and y components of a vector given its magnitude and angle.

Process of finding the resultant force vector by adding x and y components of individual vectors.

Explanation of how to determine the angle of the resultant force vector using arctangent.

Example calculation involving a westward and southward directed force vector.

Method for finding the magnitude and direction of a resultant force vector in quadrant three.

Guidance on adjusting the angle of the resultant force vector based on its quadrant.

Example of adding an eastward force vector with a force vector at an angle above the x-axis.

Detailed step-by-step process for adding vectors using their components.

Final calculation of the magnitude and direction of a resultant force vector combining various directional forces.

Transcripts

in this video we're going to talk about

how to add vectors

so a vector is a quantity that has

magnitude and direction

so let's say if we have a force vector

that's

100 newtons directed towards the east

the magnitude is the size of the force

it's 100 newtons

and the direction is east and let's say

if we want to add it to another force

that's directed east as well let's say

it's 50 newtons

the resultant sum of these two forces

is going to give us a net force of 150.

whenever you have two vectors that are

parallel to each other

you can simply add the numbers to get

the resultant sum

now let's say if we have

a 200 newton vector directed east

and 120 newton vector directed west

what is the resultant vector and also

specified the direction

so this is positive 200 because it's

directed towards the right this is

negative 120 because it's directed

towards the left

if you add these two

you should get a net force of positive

80.

now it's going to be smaller than the

original one but it's still directed

east

because this vector

is greater than this one

now as another example let's say if we

have a force vector that's 60 newtons

directed east

and another one that's 90 newtons

directed west

what is the resultant force vector

and what's the direction

so this is positive 60 that's negative

90

so we're gonna have

a net force of negative 30 newtons or

you could just say

30 newtons

directed west

now what if we have a force that's 80

newtons

directed north and another one

that's 120

direct itself

so 120 minus 8 is 40. so the net force

is going to be 40

itself

and if you recall

this is north

south east

west

now sometimes you may need to add two

vectors

that are not parallel or anti-parallel

to each other

so let's say if we have a third newton

force vector directed east

and a 40 newton force vector directed

north

what is the resultant force vector

if you have two vectors that are

perpendicular to each other

you could find the resultant force

vector

by finding the left of the hypotenuse of

the right triangle that is formed

so therefore

the resulting force vector is going to

be the square root of f1 squared plus f2

squared

let's call this f1

and this force f2

so it's going to be the square root of

30 squared

plus 40 squared

if you're familiar with the 345 triangle

then this is going to be the 30 40 50

triangle so the resultant force is going

to be 50 newtons

now sometimes in addition to finding the

magnitude of the resultant force vector

which is 50

you may need to find the direction as

well

so you got to find the angle theta

to find it you can use this formula it's

the

inverse tangent

of the

y component divided by the x component

so the force in the y direction is 40

and the x direction is 30.

so to find the angle it's going to be

our tangent

the opposite side which is the y value

of 40

over the adjacent side which is

the x value of 30.

hopefully you're familiar with sohcahtoa

sine is opposite over hypotenuse cosine

is adjacent over hypotenuse and tangent

is

opposite relative to theta over at the

adjacent side

the arc tangent of 40 over 30

is 53.1 degrees

so we can say that the resultant force

vector has a magnitude of 50 newtons and

a direction of 53.1 degrees relative to

the x-axis

let's try another example so let's say

if we have a force vector that's

50 newtons directed

west

and another one that's

120 newtons

directed

south calculate the magnitude of the

resulting force factor and find the

direction

so let's draw a triangle

so this side is 50

and this side is going to be 120

so the resultant force vector is the

hypotenuse of this triangle

so notice that it's in quadrant three

now if you're familiar with the 5 12 13

triangle

then the hypotenuse has to be 130.

so the resultant force vector is going

to be the square root of 50 squared

plus 120 squared

and that's going to turn out to be 130

newtons

so that's the magnitude of the resultant

force vector

now all we need to do

is find the angle

so first let's find a reference angle

to find a reference angle

use the arctangent formula

take the force in the y direction

divided by that in the x direction

but make it positive initially

this will give you an acute angle

between 0 and 90 and then you could

adjust it later

so our tan 120 over 50

will give you a reference angle

of 67.4

so here's the reference angle it's

inside the triangle

now what we need

is

the angle relative

to the positive x-axis

so if this is 180

then the resultant vector is 180

plus

67.4

relative to the x-axis

so it's at an angle of 247.4

so therefore

this is the resultant force vector

which has a magnitude of 130 but an

angle

of

247.4 degrees

so that's the answer

now let's say if we have a force vector

of

45 newtons directed east

and we wish to add it to a force vector

of 60 newtons directed south

find the magnitude of the resulting

force vector

and also the angle

so let's draw a triangle

so this is 45 and this is 60.

so let's find the magnitude of the

resultant force vector so it's going to

be the square root of 45 squared

plus 60 squared

which is 75 newtons

so that's the magnitude now we got to

find the angle

but let's find a reference angle first

so it's going to be arc tangent

the force in the y direction divided by

the force in the x direction so that's

60

over 45

which is 53.1

now that's the reference angle

the first force vector is positive

it's directed east

the second one is directed south

so therefore

the resultant force vector is in

quadrant four

now if this angle is 53.1

to find the angle

relative to the positive x-axis it's

going to be

360 minus 53.1 keep in mind a full

circle is 360 but you need to go back

53.1 degrees

so 360 minus 53.1

will give us an angle of 306.9

so the resultant force vector has a

magnitude

of 75 newtons

and it's directed at an angle of

306.9 degrees

relative to the positive x-axis

so this is the answer

so if you know the resultant force

vector

is going to be in quadrant one

then the angle is going to be the same

as the reference angle

and keep in mind the reference angle can

be calculated by taking the inverse

tangent of the force in the y direction

divided by the force in the x direction

and if you always make these positive

you're always going to get an acute

angle between 0 and 90 which is the

reference angle

now if you know the resultant force

vector is in quadrant two

then to find that angle it's going to be

180 minus the reference angle

if it's in quadrant three we cover this

one already it's 180 plus the reference

angle

and if you get an answer that's in

quadrant four it's going to be 360

minus the reference angle which was the

case in the last example

so

it's helpful to know these things or you

could just

see what to do visually once you graph

everything

now sometimes you may need to add

vectors

that are not parallel or perpendicular

to each other

so let's say if we want to add this

vector which is a 100 inch directed east

plus

another vector

let's say it has a magnitude of 150

but it's directed at 30 degrees

above

the x-axis

how can we find the resultant vector

first

let's re-list what we have

the first vector has a force of a

hundred

newtons

and its angle

it's east so it's on the x-axis so it

has an angle of zero degrees

the second force vector has a magnitude

of 150 newtons

and it has an angle of 30 degrees

relative to the x-axis

now

what you want to do is you want to add

these vectors using the component method

so you want to break these forces into

the components

add their respective x and y components

and then you could find the results in

force vector

so let's say if this is a force vector

this is the x component and this is the

y component

and here is the angle

f of x is equal to f

cosine theta

and f of y is f sine theta

f you can find it by using pythagorean

theorem it's f of x squared

plus

f of y squared

and the angle theta is the arc tangent

of f of y divided by f of x

so those are some formulas that you're

going to find useful

in this portion of the video

so what we need to do is find f one x

and f one y

so f one x is gonna be a hundred

cosine of zero

cosine of zero is one so a hundred times

one is simply a hundred

f one y is going to be a hundred sine of

zero sine of zero is zero a hundred

times zero

is going to be zero

now let's do the same for f2x

and f2y

f2x is 150

times cosine of 30 degrees

now if you type that in your calculator

make sure it's in

degree mode

so this will give you as a decimal

129.9 newtons

f2y that's going to be 150 times sine 30

which is just 75.

now what we need to do

is add up the x components

so if we add those two

values the sum

of the forces in the x direction is

going to be a hundred

plus

129.9

so that's going to be 229.9

and then we need to find the sum of the

forces in the y direction

which is just 0

plus 75 so that's simply

75.

so f of x

is

229.9 units and f of y

is simply

75 newtons

now let's go ahead and draw these values

on an xy plane

so f of x is 229

and f of y is 75.

the resultant force vector is the

hypotenuse

and to find it

it's going to be the square root of

f of x squared

plus f of y squared

so you should get

241.8

so that's the resulting force vector

that's the magnitude of it

now that we have the magnitude

we need to find the angle

so it's going to be

arc tangent

f of x i mean f of y which is 75

divided by f of x which is 229.9

so the angle which is already in

quadrant one

is 18.1 degrees

so now we have the magnitude and the

direction of the resultant force vector