Vectors in space

A vector is the mathematical object for anything that has a size and a direction: a displacement, a velocity, a force. What makes it different from a plain number is that adding two of them depends on how they are oriented. Walk 3 km east then 4 km north and you end up 5 km from where you started, not 7 km — the arithmetic of vectors has to encode that.

The crucial and often-missed point is that a vector has no position. Two arrows of the same length pointing the same way are the same vector, whether drawn at the origin or halfway across the page. This is exactly why a force can be "applied" anywhere and still be the same force, and why you are free to slide vectors around to place them tip-to-tail when adding them.

In coordinates a vector in space is written as an ordered triple, its components along the x, y and z axes. Those components are signed: a negative component means the vector points backwards along that axis, not that it has negative length. Addition and scalar multiplication then happen component by component, which is what turns three-dimensional geometry into arithmetic you can actually compute with. The length comes back out via Pythagoras in three dimensions, √(x² + y² + z²).