A vector has direction as well as magnitude, and that changes how they add.
Vector A quantity with both magnitude and direction.
Step 1: Something to Notice
Watch first. The explanation comes later.
A plane flying at 200 km/h with a 50 km/h crosswind travels at 206 km/h over the ground, not 250.
Where did the missing 44 km/h go?
Step 2: Find Out
Isolate the heading and note how the two components trade against each other.
Step 1 — Predict
Two 3 N forces act at right angles. What is the resultant?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
Resolve with v cos θ and v sin θ
One vector rewritten as two perpendicular ones.
Component The part of a vector along a chosen axis.
Add components, then recombine
Pythagoras for the magnitude, inverse tangent for the direction.
Perpendicular components do not interfere
A vector along one axis has no projection on the other.
The choice of axes is free
Along a slope beats horizontal and vertical, for a slope problem.
Split into two that do not interfere
Perpendicular components are independent, so a two-dimensional problem becomes two one-dimensional ones. That decomposition is the most useful single technique in mechanics.
Choose the axes to suit the problem
Axes need not be horizontal and vertical. Aligning one along an incline or along the motion often removes most of the trigonometry before any calculation starts.
Step 4: A Common Mistake
Lots of people think
“Vectors are added by adding their magnitudes.”
Step 5: Why Sailboats Can Sail Into the Wind
The same idea somewhere new.
A sailboat cannot sail directly upwind, but it can sail at around 45° to it, and by tacking back and forth it makes progress upwind overall. The reason is component thinking: the sail acts as an aerofoil, generating a force largely perpendicular to its surface, and the keel prevents the boat from moving sideways, so only the forward component of that force survives. The boat is therefore driven by a component of a force that points nowhere near where the boat is going. Nothing about it is intuitive, and everything about it falls out of resolving one vector along two chosen axes.
Step 6: Think It Through
Practice makes it stick.
Two Perpendicular Pulls
Problem 1 of 2
Two 3 N forces act at 90°. What is the resultant?
A Block on a Slope
Problem 2 of 2
Which axes make an inclined plane problem easiest?
Splitting and Combining
1 of 5
What distinguishes a vector from a scalar?
2 of 5
What is the horizontal component of a vector v at angle θ?
3 of 5
How are two vectors added?
4 of 5
When does adding magnitudes give the right answer?
5 of 5
Why can perpendicular components be treated separately?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Do two 3 N perpendicular forces give 6 N?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Explain how vectors are resolved and added, and why perpendicular components are independent.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- A vector carries direction as well as magnitude.
- Resolve with v cos θ and v sin θ, and add component by component.
- Perpendicular components are exactly independent.
- Choosing axes to suit the problem is most of the skill.