Vectors are mathematical objects that have both magnitude (size) and... Show more
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
1,005
•
28 Nov 2025
•
Comprehensive Vectors Revision Notes
GNisha
@gnisha_fhdqlrplxiup
1 / 10
Getting Started with Vectors
Think of vectors as arrows that tell you exactly where to go and how far to travel. Unlike regular numbers, vectors care about direction just as much as size.
You can name vectors using the start and end points like $\overline{AB}$ or with a single letter (usually bold or underlined). In component form, vectors look like , where the top number shows horizontal movement and the bottom shows vertical movement.
Adding and subtracting vectors works just like regular arithmetic - you simply combine the corresponding components. For example, . When multiplying by a scalar (a regular number), you multiply each component separately.
Key Point: Never try to simplify vectors like fractions - each component stays separate!
Magnitude and Position Vectors
The magnitude of a vector is its length, calculated using Pythagoras' theorem. For a 2D vector , the magnitude is . For 3D vectors, you just add the third component squared under the square root.
Position vectors tell you exactly where a point is located from the origin (0,0). If point P is at coordinates (x,y,z), then its position vector is .
Here's the crucial relationship: to find vector , you calculate (destination minus starting point). This works for any two points and is absolutely essential for exam questions.
Exam Tip: Remember that - this formula appears in virtually every vector question!
Working with Vector Examples
Let's see how these concepts work in practice. Given points A(-12, 4) and B(5, -2), you can write their position vectors as and .
To find , you calculate .
When calculating magnitudes, be extra careful with negative numbers. For the vector , the magnitude is . Notice how the negative signs disappear when you square each component.
Practice Makes Perfect: The more you practice these calculations, the more automatic they become on exam day!
Unit Vectors and Parallel Vectors
A unit vector is simply a vector with magnitude 1 - it shows pure direction without worrying about size. To create a unit vector from any vector, divide each component by the vector's magnitude.
For example, if has magnitude 5, then its unit vector is .
Parallel vectors are multiples of each other. If where k is any number, then the vectors are parallel. When k is negative, they point in opposite directions but are still parallel.
Collinearity means points lie on a straight line. If , then points A, B, and C are collinear because the vectors are parallel and share point B.
Remember: Parallel vectors can point in opposite directions - the key is that one is a scalar multiple of the other!
Section Formula and Ratio Splitting
When a point divides a line segment in a specific ratio, you can find its position using the section formula: if Q divides AB in ratio m:n, then .
However, there's an easier method that many students prefer. First, find vector . Then calculate what fraction of this vector you need based on the ratio. Finally, add this to the starting position vector.
For example, if P divides AB in ratio 1:3, then P is of the way from A to B. So , and you can find P's coordinates by adding this to A's position.
The key is understanding what the ratio actually means - if it's 1:3, then AP is 1 part while PB is 3 parts, making the total journey 4 parts.
Pro Tip: Draw a simple diagram to visualise the ratio - it makes the calculation much clearer!
i, j, k Components
The i, j, k notation is just another way to write vectors using unit vectors in each direction. Here, i points along the x-axis, j along the y-axis, and k along the z-axis.
Converting between notations is straightforward: becomes , and vice versa. If a component is missing like in $5i - 2k$, it means that component is zero.
All the same rules apply - you can add, subtract, and multiply these vectors exactly as before. For instance, .
When calculating magnitudes, convert to component form first, then use the standard formula. The notation might look different, but the mathematics stays exactly the same.
Flexibility: Being comfortable with both notations gives you options for tackling exam questions in whatever way feels clearest!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Comprehensive Vectors Revision Notes
GNisha
@gnisha_fhdqlrplxiup
Vectors are mathematical objects that have both magnitude (size) and direction, making them essential for describing movement, forces, and positions in space. This guide will take you through everything from basic vector operations to more complex concepts like unit vectors... Show more
Getting Started with Vectors
Think of vectors as arrows that tell you exactly where to go and how far to travel. Unlike regular numbers, vectors care about direction just as much as size.
You can name vectors using the start and end points like $\overline{AB}$ or with a single letter (usually bold or underlined). In component form, vectors look like , where the top number shows horizontal movement and the bottom shows vertical movement.
Adding and subtracting vectors works just like regular arithmetic - you simply combine the corresponding components. For example, . When multiplying by a scalar (a regular number), you multiply each component separately.
Key Point: Never try to simplify vectors like fractions - each component stays separate!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Magnitude and Position Vectors
The magnitude of a vector is its length, calculated using Pythagoras' theorem. For a 2D vector , the magnitude is . For 3D vectors, you just add the third component squared under the square root.
Position vectors tell you exactly where a point is located from the origin (0,0). If point P is at coordinates (x,y,z), then its position vector is .
Here's the crucial relationship: to find vector , you calculate (destination minus starting point). This works for any two points and is absolutely essential for exam questions.
Exam Tip: Remember that - this formula appears in virtually every vector question!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Working with Vector Examples
Let's see how these concepts work in practice. Given points A(-12, 4) and B(5, -2), you can write their position vectors as and .
To find , you calculate .
When calculating magnitudes, be extra careful with negative numbers. For the vector , the magnitude is . Notice how the negative signs disappear when you square each component.
Practice Makes Perfect: The more you practice these calculations, the more automatic they become on exam day!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Unit Vectors and Parallel Vectors
A unit vector is simply a vector with magnitude 1 - it shows pure direction without worrying about size. To create a unit vector from any vector, divide each component by the vector's magnitude.
For example, if has magnitude 5, then its unit vector is .
Parallel vectors are multiples of each other. If where k is any number, then the vectors are parallel. When k is negative, they point in opposite directions but are still parallel.
Collinearity means points lie on a straight line. If , then points A, B, and C are collinear because the vectors are parallel and share point B.
Remember: Parallel vectors can point in opposite directions - the key is that one is a scalar multiple of the other!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Section Formula and Ratio Splitting
When a point divides a line segment in a specific ratio, you can find its position using the section formula: if Q divides AB in ratio m:n, then .
However, there's an easier method that many students prefer. First, find vector . Then calculate what fraction of this vector you need based on the ratio. Finally, add this to the starting position vector.
For example, if P divides AB in ratio 1:3, then P is of the way from A to B. So , and you can find P's coordinates by adding this to A's position.
The key is understanding what the ratio actually means - if it's 1:3, then AP is 1 part while PB is 3 parts, making the total journey 4 parts.
Pro Tip: Draw a simple diagram to visualise the ratio - it makes the calculation much clearer!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
i, j, k Components
The i, j, k notation is just another way to write vectors using unit vectors in each direction. Here, i points along the x-axis, j along the y-axis, and k along the z-axis.
Converting between notations is straightforward: becomes , and vice versa. If a component is missing like in $5i - 2k$, it means that component is zero.
All the same rules apply - you can add, subtract, and multiply these vectors exactly as before. For instance, .
When calculating magnitudes, convert to component form first, then use the standard formula. The notation might look different, but the mathematics stays exactly the same.
Flexibility: Being comfortable with both notations gives you options for tackling exam questions in whatever way feels clearest!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
11
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Mock Exam
Quiz
Flashcards
Essay
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user