Welcome to Scalars and Vectors
Welcome to one of the most fundamental building blocks of AS Physics! Whether you are tracking the flight of a projectile, working out the tension in bridge cables, or calculating the push of an engine, understanding the difference between scalars and vectors is essential. Don't worry if resolving angles or drawing vector triangles seems tricky at first — by breaking each method down into clear, step-by-step techniques, you will quickly gain the confidence needed to score top marks in your Unit AS 1 exam.
---1. Scalars vs. Vectors: The Fundamentals
What is a Scalar Quantity?
A scalar is a physical quantity that has magnitude (size) only. It has no direction associated with it. Scalars are added, subtracted, multiplied, and divided using simple everyday arithmetic.
Standard CCEA Scalar Examples:
• Mass (e.g. \(5\text{ kg}\))
• Time (e.g. \(12\text{ s}\))
• Distance (e.g. \(100\text{ m}\))
• Speed (e.g. \(20\text{ m s}^{-1}\))
• Energy / Work Done (e.g. \(250\text{ J}\))
• Power (e.g. \(60\text{ W}\))
• Temperature (e.g. \(293\text{ K}\))
• Electric Charge (e.g. \(1.6 \times 10^{-19}\text{ C}\))
• Potential Difference / EMF (e.g. \(12\text{ V}\))
• Resistance (e.g. \(50\ \Omega\))
What is a Vector Quantity?
A vector is a physical quantity that has both magnitude (size) AND direction. Specifying only the magnitude gives an incomplete description of what is happening physically.
Standard CCEA Vector Examples:
• Displacement (e.g. \(100\text{ m}\) due North)
• Velocity (e.g. \(20\text{ m s}^{-1}\) at \(30^\circ\) to the horizontal)
• Acceleration (e.g. \(9.81\text{ m s}^{-2}\) downwards)
• Force / Weight (e.g. \(50\text{ N}\) vertically downwards)
• Linear Momentum (e.g. \(15\text{ kg m s}^{-1}\) East)
• Impulse (e.g. \(10\text{ N s}\) to the right)
• Electric / Gravitational Field Strength (e.g. \(9.81\text{ N kg}^{-1}\) towards the centre of Earth)
Memory Trick:
• Scalar = Size only.
• Vector = Value (size) + Vector direction!
Key Takeaway: A scalar tells you "how much", while a vector tells you "how much" AND "in which direction".
---2. Combining Perpendicular Vectors
When two or more vectors act simultaneously on an object, their combined effect is represented by a single vector called the resultant vector.
Definition: A resultant vector is the single vector that produces the exact same physical effect in both magnitude and direction as two or more vectors acting together.
Finding the Resultant of Two Perpendicular Vectors (\(90^\circ\) Apart)
Consider two perpendicular vectors: a horizontal vector \(V_x\) and a vertical vector \(V_y\).
Step 1: Calculate the Magnitude (\(R\)) using Pythagoras' Theorem:
\(R = \sqrt{V_x^2 + V_y^2}\)
Step 2: Calculate the Direction (\(\theta\)) using Trigonometry:
\(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{V_y}{V_x}\)
\(\theta = \tan^{-1}\left(\frac{V_y}{V_x}\right)\)
Step-by-Step Worked Example
A swimmer heads directly across a river East at \(1.5\text{ m s}^{-1}\). The river current flows South at \(0.8\text{ m s}^{-1}\). Calculate the resultant velocity of the swimmer.
1. Magnitude:
\(R = \sqrt{(1.5)^2 + (0.8)^2} = \sqrt{2.25 + 0.64} = \sqrt{2.89} = 1.7\text{ m s}^{-1}\)
2. Direction:
\(\theta = \tan^{-1}\left(\frac{0.8}{1.5}\right) = \tan^{-1}(0.5333) \approx 28.1^\circ\)
3. Full Exam Statement:
Resultant velocity = \(1.7\text{ m s}^{-1}\) at an angle of \(28.1^\circ\) South of East (or at \(28.1^\circ\) to the horizontal riverbank).
Key Takeaway: Whenever an exam question asks for a vector quantity (such as resultant force or velocity), you must provide both the magnitude and the direction relative to a clearly stated reference line.
---3. Resolving a Vector into Perpendicular Components
Resolving is the reverse process of finding a resultant. It means splitting a single vector into two independent, perpendicular parts (usually horizontal and vertical, or parallel and perpendicular to a slope).
Standard Horizontal and Vertical Components
For a vector \(V\) acting at an angle \(\theta\) to the horizontal:
• Horizontal component (\(V_x\)): \(V_x = V \cos\theta\)
• Vertical component (\(V_y\)): \(V_y = V \sin\theta\)
Note: If the angle \(\theta\) is measured relative to the vertical axis instead, the trigonometry flips: the vertical component becomes \(V_y = V \cos\theta\) and the horizontal component becomes \(V_x = V \sin\theta\).
Handy Rule of Thumb:
• The component adjacent to (touching) the angle \(\theta\) takes the \(\cos\theta\) term.
• The component opposite to (not touching) the angle \(\theta\) takes the \(\sin\theta\) term.
Resolving on an Inclined Plane (Slope)
A classic CCEA exam scenario is an object of weight \(W\) resting on a slope inclined at an angle \(\theta\) to the horizontal.
Because the angle between the vertical weight vector and the normal (perpendicular) to the slope is also \(\theta\):
• Component of weight acting down (parallel to) the slope: \(W_{\parallel} = W \sin\theta\)
• Component of weight acting into (perpendicular to) the slope: \(W_{\perp} = W \cos\theta\)
Key Takeaway: On an inclined plane of angle \(\theta\), weight pulls the object down the slope with a force equal to \(W\sin\theta\), while pressing into the slope with a force equal to \(W\cos\theta\).
---4. Scale Drawings & Graphical Methods
For non-perpendicular vectors, or where specified in an exam question, you can find the resultant using scale drawings.
Methods of Graphical Addition
1. The Tip-to-Tail Method (Triangle of Vectors):
• Draw the first vector to scale, starting from an origin.
• Place the tail of the second vector at the tip (arrowhead) of the first vector.
• The resultant vector is the straight line drawn directly from the tail of the first vector to the tip of the second vector.
2. The Parallelogram of Vectors Method:
• Draw both vectors starting from the exact same point (tail-to-tail).
• Complete the parallelogram by drawing parallel dashed lines.
• The resultant vector is the diagonal drawn from the common origin to the opposite corner.
Exam Rules for Scale Drawings
• State a clear scale: Always write down your chosen scale (e.g. \(1\text{ cm} \equiv 10\text{ N}\) or \(1\text{ cm} \equiv 2\text{ m s}^{-1}\)).
• Use sharp pencil and clear arrowheads: Every vector line must have an arrowhead indicating its direction.
• CCEA Exam Tolerances: Measurements are strictly checked against tolerances (typically within \(\pm 1\text{ to } 2\text{ mm}\) and angles within \(\pm 1^\circ\text{ to } \pm 2^\circ\)). Use a large scale that fills the page to minimise plotting errors!
Key Takeaway: In scale drawings, measure lines with a ruler to find magnitude and measure angles with a protractor to find direction.
---5. Equilibrium of Coplanar Forces
The Equilibrium Condition
An object is in translational equilibrium when the resultant force acting on it is zero:
\(\Sigma F = 0\)
This means there is no acceleration: the object remains at rest or continues to move with constant velocity.
Two Coplanar Forces in Equilibrium
For two forces to keep a point in equilibrium, they must be:
1. Equal in magnitude
2. Opposite in direction
3. Collinear (acting along the exact same straight line)
Three Coplanar Forces in Equilibrium
When three coplanar forces acting at a point are in equilibrium, two methods can be used to solve for unknown forces or angles:
Method 1: The Closed Vector Triangle
• When the three forces are placed tip-to-tail, they form a completely closed triangle.
• The arrows must follow each other continuously around the triangle in a single closed loop.
• Because you end up right back where you started, the overall resultant force is zero.
Method 2: Analytical Resolution (\(\Sigma F_x = 0\) and \(\Sigma F_y = 0\))
• Resolve all forces into horizontal (\(x\)) and vertical (\(y\)) components.
• Set the sum of all horizontal components to zero: \(\Sigma F_x = 0\) (i.e. Total Forces Left = Total Forces Right).
• Set the sum of all vertical components to zero: \(\Sigma F_y = 0\) (i.e. Total Forces Up = Total Forces Down).
• Solve the simultaneous equations for the unknown values.
Key Takeaway: If three coplanar forces are in equilibrium, their vector triangle closes completely with arrows chasing each other in a continuous loop.
---6. Common Exam Pitfalls & Examiner Tips
1. Forgetting Direction on Vector Answers:
If a question asks for "the resultant force" or "velocity", calculating the magnitude alone will lose you the final mark. Always calculate and write down the angle and the reference axis (e.g. "\(45\text{ N}\) at \(36.9^\circ\) to the horizontal").
2. Giving Vague Angles:
Writing "at an angle of \(30^\circ\)" is ambiguous. Always specify: "\(30^\circ\) above the horizontal", "\(30^\circ\) to the vertical", or provide a clear three-figure bearing (e.g. "bearing \(060^\circ\)").
3. Swapping Sine and Cosine on Slopes:
Remember: the component of weight pulling down a slope is \(W \sin\theta\), while the component pressing into the slope is \(W \cos\theta\) (where \(\theta\) is the angle of the incline to the horizontal).
4. Confusing Resultant Loops with Equilibrium Loops:
• For finding a resultant: The resultant arrow points directly from the original start point to the final end point.
• For equilibrium: The arrows chase each other continuously nose-to-tail in a closed loop.
Quick Summary Checklist
• Scalar: Magnitude only (e.g. mass, time, speed, energy, charge, resistance).
• Vector: Magnitude and direction (e.g. displacement, velocity, acceleration, force, momentum).
• Perpendicular Resultant: \(R = \sqrt{V_x^2 + V_y^2}\) and \(\theta = \tan^{-1}\left(\frac{V_y}{V_x}\right)\).
• Resolving Components: \(V_x = V\cos\theta\) and \(V_y = V\sin\theta\) (for \(\theta\) to the horizontal).
• Equilibrium: Resultant force is zero (\(\Sigma F = 0 \implies \Sigma F_x = 0\text{ and }\Sigma F_y = 0\)); three forces form a closed vector triangle.