Welcome to Scalars and Vectors!
Welcome to one of the most fundamental toolkits in A Level Physics! Whether you are tracking a rocket launch, predicting where a kicked football lands, or calculating the tension in a bridge cable, everything starts with understanding the difference between scalars and vectors.
Don't worry if you find vectors slightly confusing at first. Many students do! In this guide, we will break down every idea into clear, bite-sized steps with everyday examples and easy-to-remember tricks so you feel completely confident for your CCEA AS 1 exam.
1. Scalars vs. Vectors: The Core Difference
In physics, every measurable physical quantity can be placed into one of two families: scalars or vectors.
Scalar: A quantity that has magnitude (size) only, but no direction.
For example, if you say the temperature is \(18\text{ }^\circ\text{C}\) or a bag has a mass of \(5\text{ kg}\), the number and unit tell you everything you need to know. It makes no sense to say "\(5\text{ kg}\) to the north"!
Vector: A quantity that has both magnitude (size) and a specific direction.
For example, pushing a box with a force of \(20\text{ N}\) to the right will produce a very different result than pushing it with \(20\text{ N}\) downwards. Direction matters completely!
Quick Comparison Table of Common Quantities
Common Scalars:
• Distance (\(d\) or \(s\))
• Speed (\(v\))
• Mass (\(m\))
• Time (\(t\))
• Energy and Work done (\(E\), \(W\))
• Temperature (\(T\))
• Electric charge (\(Q\)) and Electric potential (\(V\))
Common Vectors:
• Displacement (\(s\))
• Velocity (\(u\), \(v\))
• Acceleration (\(a\))
• Force / Weight (\(F\), \(W\))
• Momentum (\(p\))
• Electric field strength (\(E\))
Analogy & Memory Aid
Memory Trick:
• Scalar = Size only.
• Vector = Value + Vector direction.
Distance vs. Displacement (A Classic Trap!)
Imagine you walk \(400\text{ m}\) around a running track and end up exactly where you started:
• Your distance travelled is \(400\text{ m}\) (scalar: total path length covered).
• Your displacement is \(0\text{ m}\) (vector: straight-line distance and direction from start to finish).
Speed vs. Velocity
A car driving in a circle at a steady \(30\text{ m s}^{-1}\) has a constant speed, but its velocity is constantly changing because its direction of travel is constantly changing!
Key Takeaway: Always ask yourself: "Does the direction change the physical outcome?" If yes, it is a vector.
2. Combining Vectors in One Dimension (Collinear Vectors)
When vectors act along the same straight line, adding them together to find the resultant vector (the single vector that has the same overall effect) is straightforward arithmetic.
Step-by-Step Method:
1. Choose a positive direction (for example, to the right or upwards as \(+\)).
2. Assign signs to each vector based on your chosen direction.
3. Add the values algebraically.
Worked Example:
Two people push a crate. Person A pushes to the right with \(30\text{ N}\). Person B pushes to the left with \(10\text{ N}\).
Taking right as positive (\(+\)):
\(F_{\text{resultant}} = (+30\text{ N}) + (-10\text{ N}) = +20\text{ N}\)
The resultant force is \(20\text{ N}\) to the right.
Key Takeaway: In 1D, vectors pointing in opposite directions subtract; vectors in the same direction add.
3. Combining Perpendicular Vectors (\(90^\circ\))
When two vectors are at right angles (\(90^\circ\)) to each other, we cannot simply add their magnitudes together. Instead, we use Pythagoras' Theorem to find the magnitude and Trigonometry to find the direction.
How to Draw Vectors: The "Tip-to-Tail" Rule
To add two vectors visually:
1. Draw the first vector as an arrow pointing in its direction.
2. Place the tail (start) of the second vector at the tip (arrowhead) of the first vector.
3. Draw the resultant vector from the tail of the very first vector straight to the tip of the second vector.
The Mathematical Formulas
For two perpendicular vectors \(A\) and \(B\):
• Magnitude of Resultant (\(R\)):
\(R = \sqrt{A^2 + B^2}\)
• Angle of Direction (\(\theta\)):
\(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{B}{A} \implies \theta = \tan^{-1}\left(\frac{B}{A}\right)\)
Worked Example: Swimmer in a River
A swimmer swims due North across a river at \(4.0\text{ m s}^{-1}\). The river current flows due East at \(3.0\text{ m s}^{-1}\). Find the resultant velocity of the swimmer.
Step 1: Calculate the magnitude using Pythagoras
\(v_{\text{resultant}} = \sqrt{(4.0\text{ m s}^{-1})^2 + (3.0\text{ m s}^{-1})^2}\)
\(v_{\text{resultant}} = \sqrt{16 + 9} = \sqrt{25} = 5.0\text{ m s}^{-1}\)
Step 2: Calculate the angle using trigonometry
Let \(\theta\) be the angle measured from the North direction towards the East:
\(\tan \theta = \frac{\text{East velocity}}{\text{North velocity}} = \frac{3.0}{4.0} = 0.75\)
\(\theta = \tan^{-1}(0.75) \approx 36.9^\circ\)
Step 3: State magnitude AND direction clearly
The resultant velocity is \(5.0\text{ m s}^{-1}\) at an angle of \(36.9^\circ\) East of North (or on a bearing of \(037^\circ\)).
Common Mistake to Avoid: Never state just the number! A vector answer is incomplete without a clear direction or reference angle.
Key Takeaway: For \(90^\circ\) vectors, use Pythagoras for magnitude and \(\tan^{-1}\) for angle.
4. Resolving a Vector into Perpendicular Components
Just as two perpendicular vectors can be combined into one resultant vector, a single vector acting at an angle can be split (or resolved) into two perpendicular components—usually horizontal (\(x\)) and vertical (\(y\)).
This is extremely useful in AS Physics because it allows us to analyze horizontal motion (like gliding forward) independently from vertical motion (like falling under gravity).
The Component Equations
Consider a vector \(F\) acting at an angle \(\theta\) to the horizontal:
• Horizontal component (\(F_x\)):
\(F_x = F \cos \theta\)
• Vertical component (\(F_y\)):
\(F_y = F \sin \theta\)
How to Always Pick Sine or Cosine Correctly
Don't just blindly memorize that "horizontal is always cosine". Instead, look at where the angle \(\theta\) sits:
• The component that is adjacent (next to) the angle gets \(\cos \theta\).
• The component that is opposite (away from) the angle gets \(\sin \theta\).
Memory Trick: "The side that is COSY (next to \(\theta\)) takes COS."
Worked Example: Pulling a Sledge
A child pulls a sledge across smooth snow with a tension force of \(50\text{ N}\) in a rope tilted at \(30^\circ\) above the ground.
1. Find the horizontal pulling force:
\(F_x = F \cos(30^\circ) = 50 \times \cos(30^\circ) = 50 \times 0.866 = 43.3\text{ N}\)
(This is the force actually causing the sledge to accelerate forwards.)
2. Find the vertical lifting force:
\(F_y = F \sin(30^\circ) = 50 \times \sin(30^\circ) = 50 \times 0.5 = 25.0\text{ N}\)
(This force slightly lifts the sledge, reducing the contact force with the snow.)
Key Takeaway: Resolving vectors allows you to break complicated diagonal motion into two simple, independent perpendicular directions.
5. Scale Drawings for Non-Perpendicular Vectors
When vectors are not at \(90^\circ\) to each other, you can find the resultant by creating an accurate scale drawing using a ruler and protractor.
How to Complete a Scale Drawing in Exam Conditions:
1. Choose a sensible scale: For example, let \(1\text{ cm} = 2\text{ N}\) or \(1\text{ cm} = 10\text{ m s}^{-1}\). Make the diagram as large as reasonably possible for maximum accuracy.
2. Draw the first vector: Use a ruler to measure the length and a protractor to set the exact angle from a drawn reference line (like a horizontal dashed line).
3. Draw the second vector: Start from the tip of the first arrow (tip-to-tail method) at the specified angle.
4. Draw the resultant line: Connect the starting tail directly to the finishing tip. Add an arrowhead to show direction.
5. Measure and convert: Measure the resultant line's length with your ruler and convert it back to physical units using your scale. Measure the angle of the resultant with your protractor.
Key Takeaway: In scale drawings, accuracy is key—use a sharp pencil, measure carefully, and always write your chosen scale clearly on the page.
6. Quick Review Checklist
Before sitting your exam on this topic, check that you can:
• Define a scalar (magnitude only) and a vector (magnitude and direction).
• Correctly categorize standard physical quantities (e.g., speed vs. velocity, distance vs. displacement).
• Combine collinear vectors using simple addition and sign conventions.
• Calculate the resultant of two perpendicular vectors using \(R = \sqrt{A^2 + B^2}\) and \(\theta = \tan^{-1}\left(\frac{\text{Opp}}{\text{Adj}}\right)\).
• Resolve any vector into perpendicular components using \(F \cos \theta\) and \(F \sin \theta\).
• Construct an accurate tip-to-tail scale diagram with a specified scale.