A vector is a quantity with size and direction—displacement, velocity, force. On paper we draw an arrow. In coordinates we write components ⟨x, y⟩: how far east and how far north from the tail to the tip.
Components and magnitude
The length (magnitude) of ⟨x, y⟩ is the straight-line distance:
|v| = √(x² + y²)
That is the Pythagorean theorem—the same 3-4-5 idea as a ladder against a wall.
Adding vectors
Add components separately: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a+c, b+d⟩. Geometrically, place the tail of the second arrow at the tip of the first—the sum is the arrow from the start of the first to the end of the second (the parallelogram rule).
Scaling
Multiply by a number k: k⟨x, y⟩ = ⟨kx, ky⟩. If |k| > 1 the arrow lengthens; if k is negative it reverses direction.
Common trap: Magnitude is never negative. Direction can reverse when you scale by a negative number, but length is |k| times the original length.
Related
- Polar coordinates — another way to store direction and distance
- Trigonometry basics