Physics (9702) | Scalars and vectors

AS & A Level Physics (9702) Study Notes: Scalars and Vectors

Hello future physicist! This chapter is your foundation. Understanding scalars and vectors isn't just about passing the test—it's about speaking the language of physics. Every calculation involving motion, force, or fields depends on correctly identifying whether direction matters. Don't worry if geometry or trigonometry feels challenging; we will break down the rules for handling these quantities step-by-step!

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1. The Core Difference: Scalars vs. Vectors (Syllabus 1.4.1)

Physics deals with quantities we can measure. We group these quantities based on whether they require a direction to be fully described.

1.1 Scalar Quantities

A scalar quantity is defined completely by its magnitude (size) only.

• It tells you "how much" but not "which way."
• When adding or subtracting scalars, you use simple arithmetic (like you did in primary school).

Memory Aid: SCALAR means Size only.

Examples of Scalars (You MUST know these!):

• Mass (e.g., 5 kg)
• Time (e.g., 10 seconds)
• Distance (e.g., 50 meters traveled)
• Speed (e.g., 20 m/s)
• Energy, Work, Power
• Temperature

1.2 Vector Quantities

A vector quantity is defined completely by both its magnitude (size) and its direction.

• It tells you "how much" AND "which way."
• Vectors are often represented graphically by arrows. The length of the arrow shows the magnitude, and the arrowhead shows the direction.
• In written formulas, vectors are often denoted by bold letters (e.g., F) or an arrow over the letter (e.g., \(\vec{F}\)).

Examples of Vectors (You MUST know these!):

• Displacement (e.g., 5 km East)
• Velocity (e.g., 15 m/s North)
• Acceleration
• Force (e.g., 10 N downwards)
• Momentum

Real-World Analogy:
Imagine you are giving instructions to a delivery driver:

• Scalar instruction: "Drive for 10 kilometres." (They could end up anywhere!)
• Vector instruction: "Drive 10 kilometres East." (This specifies exactly where they go.)

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2. Adding and Subtracting Coplanar Vectors (Syllabus 1.4.2)

Since direction is vital for vectors, you cannot just add their magnitudes together unless they are acting in the same straight line.

Coplanar simply means the vectors lie on the same 2D plane (like vectors drawn on a flat piece of paper).

2.1 Finding the Resultant Vector

When you add two or more vectors, the single vector that has the same effect is called the resultant vector (\(\mathbf{R}\)).

2.2 Case 1: Vectors on the Same Straight Line

• Same Direction: Add magnitudes. (Example: 5 N East + 3 N East = 8 N East).
• Opposite Direction: Subtract magnitudes. The resultant direction is that of the larger vector. (Example: 5 N East + 3 N West = 2 N East).

2.3 Case 2: Graphical Addition (Non-Parallel Vectors)

The simplest way to visualize non-parallel vector addition is using the Triangle Rule (or Head-to-Tail method).

Step-by-Step Triangle Rule:

1. Draw the first vector (\(\mathbf{A}\)) to scale and in the correct direction.
2. Draw the second vector (\(\mathbf{B}\)) starting from the head (tip) of the first vector.
3. The resultant vector (\(\mathbf{R}\)) is drawn from the tail (start) of the first vector to the head of the second vector.

Imagine following a treasure map: Vector A is the first instruction, Vector B is the second. The resultant R is the straight line path from start to finish.

2.4 Case 3: Subtraction of Vectors

Subtracting a vector is the same as adding its negative.

\( \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}) \)

The vector \( (-\mathbf{B}) \) has the same magnitude as \(\mathbf{B}\) but points in the exact opposite direction (180° difference).

Process: To calculate \(\mathbf{A} - \mathbf{B}\), simply reverse the direction of \(\mathbf{B}\) and then use the standard graphical addition (Triangle Rule) with \(\mathbf{A}\) and \(-\mathbf{B}\).

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3. Resolving Vectors into Perpendicular Components (Syllabus 1.4.3)

This is arguably the most powerful technique in vector physics, as it allows us to turn complex 2D vector problems into two separate, simple 1D problems.

3.1 The Concept of Components

Any vector (\(\mathbf{F}\)) acting at an angle can be thought of as being made up of two smaller, independent vectors acting at right angles to each other (usually horizontally along the x-axis, \(F_x\), and vertically along the y-axis, \(F_y\)).

Think of pushing a lawnmower: you are pushing downwards and forwards simultaneously. The component acting downwards pushes the blade into the grass, while the component acting forwards causes the mower to move.

3.2 Calculating the Components

We use basic trigonometry (SOH CAH TOA) to find the magnitude of these components. Assume the vector \(\mathbf{F}\) has magnitude \(F\) and makes an angle \(\theta\) with the horizontal (x-axis).

1. The Component Adjacent to the Angle (Usually Horizontal, \(F_x\)):

Using Cosine (CAH: Cosine = Adjacent / Hypotenuse):
\( \cos \theta = \frac{F_x}{F} \)

\( F_x = F \cos \theta \)

2. The Component Opposite the Angle (Usually Vertical, \(F_y\)):

Using Sine (SOH: Sine = Opposite / Hypotenuse):
\( \sin \theta = \frac{F_y}{F} \)

\( F_y = F \sin \theta \)

Remember: These two components must always form a right-angled triangle with the original vector F as the hypotenuse.

3.3 Step-by-Step Example of Resolution

A force of 100 N is applied to a box at an angle of 30° above the horizontal.

Step 1: Identify the angle and the magnitude.
\( F = 100 \, \text{N} \)
\( \theta = 30^\circ \)

Step 2: Calculate the horizontal component (\(F_x\)). (This is the force that moves the box forward.)
\( F_x = F \cos \theta \)
\( F_x = 100 \times \cos(30^\circ) \)
\( F_x \approx 86.6 \, \text{N} \)

Step 3: Calculate the vertical component (\(F_y\)). (This is the force that lifts the box slightly.)
\( F_y = F \sin \theta \)
\( F_y = 100 \times \sin(30^\circ) \)
\( F_y = 50.0 \, \text{N} \)

Key Takeaway: The single 100 N force has the same effect as applying a horizontal force of 86.6 N and a vertical force of 50.0 N simultaneously.

3.4 Recombining Components to Find the Resultant (Pythagoras)

If you have multiple forces resolved into their separate x and y components, you can find the final resultant vector (\(\mathbf{R}\)) by:

1. Summing all the x-components to get the total horizontal component (\(\Sigma F_x\)).
2. Summing all the y-components to get the total vertical component (\(\Sigma F_y\)).
3. Using Pythagoras' Theorem to find the resultant magnitude (\(R\)):
   \( R = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2} \)
4. Using Trigonometry to find the direction (\(\phi\)):
   \( \tan \phi = \frac{\Sigma F_y}{\Sigma F_x} \)

Key Takeaway for Section 3: Resolving a vector means breaking it into perpendicular components, usually x and y, using \( F \cos \theta \) and \( F \sin \theta \). This simplifies complex problems by treating horizontal and vertical motions separately.

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Chapter Summary: Scalars and Vectors

This chapter provides the essential tools for describing all physical interactions. You must be confident in:

• Distinguishing between scalars (magnitude only, simple arithmetic) and vectors (magnitude and direction, requires geometry).
• Adding vectors graphically (Head-to-Tail rule).
• The critical skill of resolving vectors into perpendicular components (\(F \cos \theta\) and \(F \sin \theta\)) to manage 2D motion and forces effectively.

Keep practicing vector resolution—it is the most important calculation skill you will carry forward into Kinematics and Dynamics!