Vectors as Space Arrows
Before you learn a single symbol, build the picture in your head: a vector is just an arrow in space, pointing somewhere with some length.
Intuition First
Imagine standing in an empty, featureless space. If someone tells you to "move 5 units," the instruction is meaningless — 5 units in which direction? A vector exists precisely to remove that ambiguity: it's an arrow anchored in space that encodes two pieces of information at once — magnitude (how long) and direction (which way it points).
A scalar, by contrast, is just a lonely number — temperature, mass, price. A vector is an active instruction: a push, a displacement, a velocity. It doesn't just say "how much," it says "which way."
Start With a Number You Can Check by Hand
Before any general formula, look at one concrete arrow: starting at the origin, walk 3 steps right and 4 steps up.
That's just the Pythagorean theorem — the arrow's length is the hypotenuse of the right triangle formed by its horizontal and vertical steps. Keep this "3-4-5" picture in mind; the general formula below is just this idea stretched to any number of dimensions.
Try It — Vector Addition
Drag the tip of either arrow. Watch how they join tip-to-tail to form a new vector, v + w.
Formal Definition
In an n-dimensional real space ℝⁿ, a vector v is written as an ordered list of numbers:
Its magnitude is exactly the 3-4-5 idea above, generalized — square every component, add them up, take the root: