Define the 'Resultant Vector'.

The resultant vector is a single vector that has the same effect as two or more vectors acting together. It represents the vector sum of all individual components in the system.

What is an 'Inverse Vector' in the context of subtraction?

An inverse vector has the same magnitude as the original vector but points in the exactly opposite direction (180 degrees away). It is denoted as $-\vec{V}$.

What is the primary difference between the Triangle Method and the Parallelogram Method of vector addition?

The Triangle Method connects vectors head-to-tail to show a sequence of events, while the Parallelogram Method connects them tail-to-tail to show vectors acting simultaneously from a single point. Both yield the same resultant vector.

How does vector addition differ from scalar addition?

Scalar addition only considers magnitude (e.g., $5kg + 5kg = 10kg$), whereas vector addition must account for direction. Two $5N$ forces could result in any value from $0N$ to $10N$ depending on the angle between them.

When should you use Pythagoras' Theorem instead of a scale drawing to find a resultant?

Pythagoras' Theorem should be used when the two vectors are perpendicular (at a 90-degree angle) to each other. It provides an exact mathematical answer, whereas scale drawings are subject to measurement errors.

What is a common error when calculating the magnitude of a resultant for two perpendicular vectors?

A common error is simply adding the two magnitudes together (e.g., $3 + 4 = 7$) instead of using the square root of the sum of squares ($\sqrt{3^2 + 4^2} = 5$). This ignores the geometric relationship of the vectors.

What mistake is often made when performing vector subtraction graphically?

Students often forget to reverse the direction of the vector being subtracted. To find $\vec{A} - \vec{B}$, you must flip $\vec{B}$ to point in the opposite direction before adding it to $\vec{A}$.

Why is it a mistake to measure the angle of a resultant without specifying a reference line?

An angle like '30 degrees' is ambiguous. You must specify if it is from the horizontal, the vertical, or a compass bearing (e.g., North) so the direction is uniquely defined.

What does it mean for a set of vectors to be in 'Equilibrium'?

Equilibrium occurs when the resultant of all combined vectors is zero. Graphically, this is shown when the vectors form a closed loop, ending exactly where the first one started.

Why can the magnitude of a resultant never be greater than the sum of the individual magnitudes?

The shortest distance between two points is a straight line. Any angle between vectors (other than 0 degrees) creates a triangle where the third side (resultant) must be shorter than the sum of the other two sides.

Combining Vectors | OCR AS-Level Physics

2. Foundations of Physics

Combining Vectors

Summary

Combining vectors involves determining a single 'resultant' vector that represents the cumulative effect of two or more individual vectors. Unlike scalars, vectors possess both magnitude and direction, requiring geometric or trigonometric methods to find their sum or difference.

1. Definition & Core Concepts

Vector Quantities: These are physical quantities, such as velocity or force, defined by both a numerical magnitude and a specific direction.
Resultant Vector: Often referred to as the 'net' vector, the resultant is the single vector that produces the same effect as all the original vectors combined.
Graphical Representation: Vectors are drawn as arrows where the length represents the magnitude (to scale) and the arrowhead points in the direction of the quantity.

2. Geometric Addition Methods

Triangle Method (Tip-to-Tail): To add two vectors, place the tail of the second vector at the head (tip) of the first. The resultant vector is drawn from the tail of the first to the head of the second.
Parallelogram Method (Tail-to-Tail): Place both vectors so their tails meet at the same origin. Complete a parallelogram using these vectors as adjacent sides; the resultant is the diagonal starting from the common origin.
Polygon Method: For more than two vectors, continue the tip-to-tail sequence for all vectors. The resultant connects the very first tail to the very last head.

Diagram showing the triangle method of vector addition where Vector B is placed at the tip of Vector A, and the resultant connects the start of A to the end of B.

3. Mathematical Calculation (Right-Angled)

4. Scale Drawing (Non-Right-Angled)

Scale Selection: Choose a conversion factor (e.g., 1 cm = 10 Newtons) that allows the diagram to fit on the page while remaining large enough for accuracy.
Measurement Tools: Use a ruler to draw lengths proportional to magnitudes and a protractor to set the correct angles relative to a reference line (like North or the horizontal).
Final Conversion: After drawing the resultant, measure its length with a ruler and multiply by the scale factor to find the real-world magnitude.

5. Vector Subtraction

6. Key Distinctions

7. Exam Strategy & Tips