Vectors.
How much, and which way?
“Walk 3 metres” leaves a question unanswered. Walk where? A vector gives an amount and a direction together.
No previous physics or trigonometry needed. Start with an arrow, then build a journey.
An arrow tells two things.
Its length represents the magnitude (the amount). Its arrowhead shows the direction. Here every grid square represents 1 metre of displacement.
Drag the large circle at the arrowhead, or use the sliders. The faint arrow is its original position.
Moving this displacement arrow leaves its magnitude and direction unchanged.
A: first walk + B: second walk → R: resultant
Make a vector of magnitude 5 m, pointing northeast, with components 3 m east and 4 m north.
A number such as 3 m is a magnitude. The vector also needs the direction.
Distance ≠ displacement.
Distance counts the total length of the route you travel. It is a scalar: an amount without a direction.
Displacement points from where you started to where you ended. It is a vector. Different routes can have the same displacement.
On a return trip to your starting point, your displacement is the zero vector. Its magnitude is zero and it has no defined direction.
Out and back.
You walk 2 m east, then 2 m west. What is your final displacement?
Where do vectors appear in mechanics?
Displacement, velocity, acceleration and force are vectors. Distance, speed, time and mass are scalars. Each physical quantity has its own units; add vectors representing the same kind of quantity, in compatible units.
Here we move arrows to compare displacements. This freedom does not mean you can move a force anywhere on a solid object without changing its turning effect.
Components, angles and the model
This is a flat, two-dimensional displacement model. Right is east (positive x), up is north (positive y). Components are signed distances in those directions. The arrow length is √(x² + y²). An angle, when shown, is measured counterclockwise from east. Values are rounded for display.
The animated walker follows the chosen route; animation speed is for explanation and is not a measured physical speed. Components can be adjusted with a keyboard, mouse or touch. With reduced motion enabled, the walk advances one leg per click.
Sources and further reading
OpenStax University Physics: Scalars and Vectors · Coordinate Systems and Components.