QCE Physics - Unit 3 - Gravity and motion

Vector Analysis and Resultants | QCE Physics

Learn QCE Physics vector analysis with component resolution, vector addition, vector subtraction and resultant magnitude and direction.

Updated 2026-06-15 - 5 min read

Vectors are quantities with both size and direction. In QCE Physics, vectors are the language underneath projectile motion, inclined planes, circular motion, electric fields and magnetic forces. If a quantity points somewhere, you usually need to resolve it into components before you can calculate with it cleanly.

Scalars, vectors and directions

A scalar has magnitude only. Mass, time, distance, speed, energy and temperature are scalars. A vector has magnitude and direction. Displacement, velocity, acceleration, force, electric field strength and magnetic force are vectors.

The direction matters because two vectors with the same magnitude can have different effects. A $20\ \mathrm{N}$ force to the right and a $20\ \mathrm{N}$ force upward are not interchangeable. They act on different axes.

For most Unit 3 problems, choose perpendicular axes and keep a sign convention:

• right and upward are often positive
• left and downward are often negative
• north/east axes can be used for navigation-style displacement questions
• axes can be tilted to match an inclined plane

The best axes are the ones that make the physics easier. For a projectile, horizontal and vertical axes are natural. For a ramp, axes parallel and perpendicular to the slope are usually better than horizontal and vertical axes.

Resolving a vector into components

If a vector has magnitude $A$ and makes an angle $\theta$ above the positive horizontal axis, its components are:

$ A_x = A\cos\theta $

$ A_y = A\sin\theta $

This is not a formula to memorise blindly. The cosine component is adjacent to the angle, and the sine component is opposite the angle. If the angle is measured from the vertical instead, the sine and cosine roles swap. A quick sketch prevents most component errors.

Components must also carry signs. A vector pointing down and right has a positive $x$ component and a negative $y$ component. A vector pointing up and left has a negative $x$ component and a positive $y$ component.

Adding and subtracting vectors

To add vectors analytically:

1. Resolve each vector into perpendicular components.
2. Add all $x$ components to get $R_x$.
3. Add all $y$ components to get $R_y$.
4. Recombine using Pythagoras and trigonometry.

The resultant magnitude is:

$ R = \sqrt{R_x^2 + R_y^2} $

The direction can be found with:

$ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) $

The calculator angle alone may not tell the full direction. If $R_x$ is negative or $R_y$ is negative, check the quadrant and express the answer clearly, such as "$36^\circ$ north of west" or "$24^\circ$ below the positive horizontal".

Subtraction is handled the same way, except subtract matching components:

$ \vec{A}-\vec{B}=(A_x-B_x)\hat{i}+(A_y-B_y)\hat{j} $

This is useful for relative velocity and displacement changes, even when the syllabus point only says "adding or subtracting components".

Why this matters for forces

Forces combine by vector addition because the net force is the resultant force:

$ \vec{F}_{net}=\sum \vec{F} $

If several forces act on an object, the object accelerates in the direction of the net force, not necessarily in the direction of any one force. This is why free-body diagrams are so important. A normal force, weight, friction and tension may point in different directions; the object responds to the combined result.

In an inclined-plane question, resolving weight into components lets you separate the part of gravity pulling the object down the slope from the part pressing it into the surface. In a projectile question, resolving velocity lets you treat horizontal and vertical motion independently.

Worked example

Exam-ready method

Set out vector work in a table when there is more than one vector. Use columns for magnitude, angle, $x$ component and $y$ component. This makes sign errors visible and gives a marker a clear path through your reasoning.

If the answer is a force, include newtons. If it is a velocity, include $\mathrm{m\ s^{-1}}$. If it is a displacement, include metres and a direction. A vector answer without a direction is incomplete unless the question only asks for magnitude.

Quick check

Sources

• QCAA Physics subject page
• QCAA Physics 2025 syllabus
• QCAA Physics 2025 formula and data book
• OpenStax University Physics Volume 1: Vectors
• OpenStax University Physics Volume 1: Vector Addition and Subtraction