Welcome to the World of Vectors!
Hi there! Today we are diving into one of the most useful tools in your mathematical toolkit: Vectors. Whether you’re planning the flight path of a plane, programming a video game, or understanding how forces push on a bridge, vectors are the language you’ll use.
In this guide, we will break down what vectors are, how to draw them, and how to use them to solve real-world problems. Don't worry if it seems a bit "abstract" at first—by the end of these notes, you’ll be seeing vectors everywhere!
1. Scalar vs. Vector: What’s the Difference?
Before we start, we need to know what we are dealing with. In math, quantities come in two "flavors":
- Scalars: These are just numbers. They tell us "how much." Examples: Mass (5kg), Time (10 seconds), Temperature (25°C), or Speed (20 m/s).
- Vectors: These tell us "how much" AND "in what direction." Examples: Velocity (20 m/s North), Displacement (5 km East), or Force (10 Newtons downwards).
Analogy: Imagine you are lost in a desert. If I tell you "The water is 5km away," that is a scalar (distance). You might walk 5km in the wrong direction! If I say "The water is 5km North," that is a vector (displacement). Now you know exactly where to go.
Quick Review: A vector has both magnitude (size) and direction.
2. How We Write and Draw Vectors
Because vectors have direction, we can't just write them as normal numbers. We use special notation:
Notation
- Bold letters: In textbooks, you’ll see vectors written as a or b.
- Underlined letters: Since you can't write in bold with a pen, you should always underline your vectors, like this: \(\underline{a}\). Common Mistake: Forgetting the underline! This tells the examiner you know it's a vector, not just a number.
- From Point to Point: If a vector goes from point A to point B, we write it as \(\vec{AB}\).
Unit Vectors: \(\mathbf{i}\) and \(\mathbf{j}\)
We use two "standard" building blocks to describe any 2D vector:
- \(\mathbf{i}\): A vector of length 1 unit in the positive x-direction (right).
- \(\mathbf{j}\): A vector of length 1 unit in the positive y-direction (up).
For example, a vector v = \(3\mathbf{i} + 4\mathbf{j}\) means "go 3 steps right and 4 steps up."
Column Vectors
Another way to write the same vector is in a bracket: \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\). The top number is the x-movement (\(\mathbf{i}\)) and the bottom number is the y-movement (\(\mathbf{j}\)).
3. Magnitude and Direction
Sometimes we know the steps (3 right, 4 up), but we want to know the direct distance (the hypotenuse) and the angle.
Finding Magnitude (Length)
The magnitude of a vector \(\mathbf{a} = x\mathbf{i} + y\mathbf{j}\) is written as \(|\mathbf{a}|\). We use Pythagoras' Theorem to find it:
\(|\mathbf{a}| = \sqrt{x^2 + y^2}\)
Example: For \(\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}\), the magnitude is \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
Finding Direction (Angle)
To find the angle \(\theta\) the vector makes with the positive x-axis, we use trigonometry:
\(\tan(\theta) = \frac{y}{x}\), so \(\theta = \tan^{-1}(\frac{y}{x})\)
Don't forget: Always draw a quick sketch! If your vector is \(-3\mathbf{i} + 4\mathbf{j}\), it’s pointing into the second quadrant (left and up), so your calculator might give you a negative angle that you'll need to adjust.
Key Takeaway: Magnitude is the "length" of the arrow, and Direction is the "angle" it points.
4. Adding and Subtracting Vectors
This is much simpler than it looks! To add vectors, you just add their components.
The Math Way
If \(\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\):
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2+3 \\ 5+(-1) \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}\)
The Visual Way (The Resultant)
Adding two vectors is like taking a journey. If you follow vector a, and then follow vector b from where a ended, the "shortcut" from the very start to the very end is the Resultant vector (\(\mathbf{a} + \mathbf{b}\)).
Memory Aid: "Head-to-Tail" method. Put the tail of the second vector on the head (arrow tip) of the first one.
5. Resolving Vectors
This is "un-doing" a vector. If you have a force of 10N at an angle of 30°, how much is pulling right and how much is pulling up?
- Horizontal component: \(F_x = F \cos(\theta)\)
- Vertical component: \(F_y = F \sin(\theta)\)
Trick: If the component is "CO"-se to the angle, use COS. (e.g., the component touching the angle \(\theta\) uses cos).
6. Vectors in Mechanics (Applying what you've learned)
In your M1 exam, you'll apply these to physical situations. The most common ones are:
Velocity and Displacement
If a particle starts at a position \(\mathbf{r_0}\) and moves with a constant velocity \(\mathbf{v}\) for time \(t\), its new position \(\mathbf{r}\) is:
\(\mathbf{r} = \mathbf{r_0} + \mathbf{v}t\)
Example: A boat starts at \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\) and moves with velocity \(3\mathbf{i} + 2\mathbf{j}\) km/h. Where is it after 2 hours?
Answer: \(\mathbf{r} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} + 2 \times \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \end{pmatrix}\).
Forces and Acceleration
Newton's Second Law (\(F=ma\)) works perfectly with vectors too!
\(\sum \mathbf{F} = m\mathbf{a}\)
If multiple forces are acting on an object, you add them all up (find the resultant force) to find the acceleration vector.
Did you know? If the resultant force is zero, the object is in equilibrium. This means it is either perfectly still or moving at a constant speed in a straight line!
Summary Checklist: Are you exam-ready?
- Can you calculate the magnitude of a vector using Pythagoras?
- Do you remember to underline your vectors (\(\underline{u}\))?
- Can you find the angle using \(\tan^{-1}\) and a sketch?
- Can you add vectors by adding the \(\mathbf{i}\) and \(\mathbf{j}\) parts separately?
- Do you know how to resolve a vector into \(F \cos(\theta)\) and \(F \sin(\theta)\)?
Final Tip: Don't let the notation scare you. At the end of the day, vectors are just instructions for moving from point A to point B. You've got this!