Welcome to the World of Vectors!
Hi there! Welcome to your study notes for Vectors in Mechanics. If you’ve ever followed directions to a friend's house or watched a plane fly through a crosswind, you’ve already encountered vectors in real life. In this chapter, we are going to learn how to use vectors as a mathematical "language" to describe how things move and the forces acting on them. Don't worry if this seems a bit abstract at first—we’ll break it down into simple, manageable steps!
Why is this important? In Mechanics, most things (like how fast you are going or which way you are being pushed) have a direction. Vectors allow us to keep track of both "how much" and "which way" all at once.
1. The Basics: What is a Vector?
Before we jump into the math, let's look at the difference between two key types of measurements:
• Scalars: These only have magnitude (size). Examples include time (5 seconds), mass (10 kg), and distance (100 meters). They don't have a direction.
• Vectors: These have both magnitude AND direction. Examples include displacement (10 meters North), velocity (5 m/s East), and force (10 Newtons downwards).
Notation (How we write them)
In your exam, vectors are usually written in bold (like a) or underlined if you are writing by hand (like a). There are two main ways to write a vector in 2D:
1. Component Form: Using unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). \( \mathbf{i} \) is one unit in the positive \( x \)-direction, and \( \mathbf{j} \) is one unit in the positive \( y \)-direction.
Example: \( \mathbf{v} = 3\mathbf{i} + 4\mathbf{j} \)
2. Column Form: Written as \( \begin{pmatrix} x \\ y \end{pmatrix} \).
Example: \( \mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \)
Quick Tip: Think of \( \mathbf{i} \) and \( \mathbf{j} \) as instructions on a map. "\( 3\mathbf{i} + 4\mathbf{j} \)" just means "go 3 steps East and 4 steps North."
Key Takeaway: A vector tells you a size and a specific direction. Always look for the \( \mathbf{i} \) and \( \mathbf{j} \) components!
2. Magnitude and Direction
Sometimes you’ll have the components (\( \mathbf{i} \) and \( \mathbf{j} \)) but you need to know the total length of the vector or the angle it makes with the ground.
Finding the Magnitude (The "How Big" part)
The magnitude of a vector \( \mathbf{a} = x\mathbf{i} + y\mathbf{j} \) is written as \( |\mathbf{a}| \). To find it, we use Pythagoras' Theorem:
\( |\mathbf{a}| = \sqrt{x^2 + y^2} \)
Finding the Direction (The "Which Way" part)
We usually describe the direction as an angle (\( \theta \)) from the positive \( x \)-axis (the \( \mathbf{i} \) direction). We use trigonometry:
\( \tan \theta = \frac{y}{x} \)
Did you know? This is exactly how GPS works! It calculates your "position vector" relative to satellites to figure out where you are on the planet.
Common Mistake to Avoid: When calculating the angle, always draw a quick sketch of the vector! If your vector is \( -3\mathbf{i} + 2\mathbf{j} \), it's pointing to the left and up. Your calculator might give you a negative angle, so use your sketch to find the correct bearing or angle from the axis required.
Key Takeaway: Magnitude is the hypotenuse of a right-angled triangle. Direction is the angle found using \( \tan \).
3. Resultants and Resolving
Mechanics is often about combining different things or breaking them down to make them easier to handle.
The Resultant Vector
The resultant is just a fancy word for the "total" or "sum." If two forces \( \mathbf{F}_1 = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{F}_2 = 4\mathbf{i} - 1\mathbf{j} \) act on an object, the resultant force \( \mathbf{R} \) is:
\( \mathbf{R} = \mathbf{F}_1 + \mathbf{F}_2 = (2+4)\mathbf{i} + (3-1)\mathbf{j} = 6\mathbf{i} + 2\mathbf{j} \)
Resolving a Vector
This is the opposite of finding the magnitude. If you know the magnitude \( R \) and the angle \( \theta \), you can find the \( \mathbf{i} \) and \( \mathbf{j} \) parts:
• Horizontal component (\( \mathbf{i} \)) = \( R \cos \theta \)
• Vertical component (\( \mathbf{j} \)) = \( R \sin \theta \)
Memory Aid: COS is "COS-close" to the angle. If the component is touching the angle \( \theta \), use \( \cos \). If it's opposite the angle, use \( \sin \).
Key Takeaway: Add components to find the resultant. Use \( \sin \) and \( \cos \) to break a diagonal vector into horizontal and vertical parts.
4. Vectors in Kinematics (Motion)
In Unit M1, we apply vectors to displacement, velocity, and acceleration. The beauty of vectors is that they work just like the numbers you used in GCSE, but for two directions at once!
Constant Velocity
If an object moves with constant velocity \( \mathbf{v} \), its displacement \( \mathbf{s} \) after time \( t \) is:
\( \mathbf{s} = \mathbf{v}t \)
If the object starts at a position vector \( \mathbf{r}_0 \), its position \( \mathbf{r} \) at any time \( t \) is:
\( \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t \)
Constant Acceleration
When acceleration \( \mathbf{a} \) is constant, we use the vector version of the SUVAT equations:
• \( \mathbf{v} = \mathbf{u} + \mathbf{a}t \)
• \( \mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2 \)
Note: You cannot use \( v^2 = u^2 + 2as \) in vector form because you can't "square" a vector directly in this context!
Analogy: Imagine walking on a moving walkway at the airport. Your "position" is where you started (\( \mathbf{r}_0 \)) plus the speed of the walkway multiplied by time (\( \mathbf{v}t \)).
Quick Review Box:
• Displacement (\( \mathbf{s} \)): Change in position vector.
• Velocity (\( \mathbf{v} \)): Rate of change of displacement.
• Acceleration (\( \mathbf{a} \)): Rate of change of velocity.
Key Takeaway: Use \( \mathbf{r} = \mathbf{r}_0 + \mathbf{v}t \) for constant velocity problems (like ships sailing at a steady speed).
5. Forces as Vectors (Dynamics)
Forces are vectors because it matters which way you push! Newton's Second Law (\( F = ma \)) works perfectly with vectors.
The Formula
\( \sum \mathbf{F} = m\mathbf{a} \)
This means if you add up all the force vectors (the resultant force), it equals the mass (a scalar) times the acceleration vector.
Equilibrium
If a particle is in equilibrium, it means it is not accelerating (it's either still or moving at a constant velocity). In vector terms:
Resultant Force \( = 0\mathbf{i} + 0\mathbf{j} \)
This means the sum of the \( \mathbf{i} \) components is 0 AND the sum of the \( \mathbf{j} \) components is 0.
Example: If forces \( \begin{pmatrix} 2 \\ p \end{pmatrix} \) and \( \begin{pmatrix} q \\ -5 \end{pmatrix} \) are in equilibrium, then:
\( 2 + q = 0 \rightarrow q = -2 \)
\( p - 5 = 0 \rightarrow p = 5 \)
Key Takeaway: Force vectors tell you the direction of acceleration. In equilibrium, all components must cancel out to zero.
Final Summary of Key Terms
• Unit Vector: A vector with magnitude 1 (like \( \mathbf{i} \) and \( \mathbf{j} \)).
• Position Vector: A vector from the origin \( (0,0) \) to the object's location.
• Speed: The magnitude of the velocity vector (\( |\mathbf{v}| \)). Speed is a scalar!
• Distance: The magnitude of the displacement vector (or the total path length).
• Resultant: The single vector that has the same effect as all the original vectors combined.
Congratulations! You've just covered the essentials of Vectors in Mechanics. Keep practicing by breaking every vector down into its \( \mathbf{i} \) and \( \mathbf{j} \) components—it makes the hard problems much easier to solve!