Build a point from directions
Can two selected vectors reach every point in the plane?
Entering midway? A vector records a direction and magnitude using ordered coordinates. A scalar multiplies every coordinate.
Start with a concrete question
Combine two arrows, then replace one by a parallel arrow and inspect what becomes unreachable.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict before calculating: Can two selected vectors reach every point in the plane?
Change one thing. Inspect what happens.
- Construct the initial endpoint and calculate both coordinates.
- Switch to the slanted basis while keeping coefficients fixed.
- Switch to parallel vectors and try to reach y=1.
- Explain independence, span and unique coordinates with a fresh example.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Vector combination on equal coordinate scales
Horizontal: x coordinate. Vertical: y coordinate.
- 1. a times u (solid line)
- 2. b times w from first endpoint (dashed line)
- 3. result vector (dashed line)
- 4. result endpoint (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x coordinate | y coordinate |
|---|---|---|
| a times u | 0 | 0 |
| a times u | 2 | 0 |
| b times w from first endpoint | 2 | 0 |
| b times w from first endpoint | 2 | 1 |
| result vector | 0 | 0 |
| result vector | 2 | 1 |
| result endpoint | 2 | 1 |
Follow the calculation
- u=(1,0); w=(0,1); coefficients a=2, b=1.
- a u + b w = (2 + 1×0, 0 + 1×1) = (2,1).
- Dot product of result with itself: 2² + 1² = 5.
- These two vectors are independent and span the plane: each point has unique coefficients.
Coordinates depend on the chosen basis. A pair of vectors is not a basis of the plane merely because it contains two vectors.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Build a point from directions
Combine two arrows, then replace one by a parallel arrow and inspect what becomes unreachable.
Concepts to recall
A vector records a direction and magnitude using ordered coordinates. A scalar multiplies every coordinate.
Understand the mechanism
With u=(1,0), w=(0,1), coefficients 2 and 1 give (2,1).
With w=(1,1), the same coefficients give (3,1); coefficients describe a basis, not an absolute point by themselves.
Parallel directions span only a line. A basis of the plane needs two independent directions. The squared length of (x,y) is its dot product with itself, x²+y².
Symbols and calculation
a u+b w=(a u_x+b w_x, a u_y+b w_y). a,b are scalars; u,w are two-coordinate column vectors.
Worked example
u=(1,0); w=(0,1); coefficients a=2, b=1.
a u + b w = (2 + 1×0, 0 + 1×1) = (2,1).
Dot product of result with itself: 2² + 1² = 5.
These two vectors are independent and span the plane: each point has unique coefficients.
Your investigation
Construct the initial endpoint and calculate both coordinates.
Switch to the slanted basis while keeping coefficients fixed.
Switch to parallel vectors and try to reach y=1.
Explain independence, span and unique coordinates with a fresh example.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
For u=(1,0), w=(1,2), a=3 and b=2, what is the resulting y coordinate?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
Workplace investigation
Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.