A point tells you where; a vector tells you which way
On the coordinate rung you gave every point an address: an ordered pair (x, y) in the plane, and you can already feel the (x, y, z) waiting in space. A point is a place. But geometry is full of things that are not places — a push, a displacement, a velocity, the step that carries you from one corner of a room to another. These have a direction and a size but no fixed home; the same shove is the same shove wherever you stand. The object that captures exactly 'a direction together with a magnitude' is a vector.
Draw a vector as an arrow. The arrow's length is its magnitude (how much), the way it points is its direction (which way). Here is the liberating part: two arrows that have the same length and the same direction are the same vector, even if they are drawn in different corners of the page. A vector is not nailed to one spot the way a point is — you are free to slide it around, keeping length and direction fixed, until it starts wherever it is convenient. That freedom is what makes vectors so easy to add.
From arrow to numbers: components
An arrow is vivid but hard to compute with; the coordinate idea rescues us again. Slide the vector so its tail sits at the origin O; then its head lands on some point, and the coordinates of that head are the vector's components. In the plane a vector v has two of them, v = (a, b); in space it has three, v = (a, b, c). The component a says how far the arrow reaches along the x-direction, b along y, c along z — the arrow's single motion split into its pure horizontal, depth, and vertical parts. These are the components, and from here on a vector simply is its list of components.
What if the arrow does not start at the origin? Say it runs from a point A to a point B; we write that vector AB. To find its components, you subtract — head minus tail: AB = (B_x - A_x, B_y - A_y, B_z - A_z). The subtraction quietly slides the arrow back to the origin for you. A special, important case is when the tail is the origin itself: then the vector from O to a point P has components equal to P's own coordinates, and we call it the position vector of P. This is the bridge between the two worlds — every point P comes with a vector OP, and that is how 'where' and 'which way' shake hands.
A = (1, 2, 3) B = (4, 6, 3) AB = B - A = (4-1, 6-2, 3-3) = (3, 4, 0) head minus tail -> the arrow from A to B, slid back to O
Adding arrows: tip to tail
Here is the move the title promises. To add two vectors u and v, draw u, then start v at the tip of u; the arrow from u's tail to v's tip is the sum u + v. This tip-to-tail rule is just 'do one displacement, then the other' — walk three steps east, then four steps north, and the single arrow from start to finish is the sum. Slide the two arrows the other way around (v first, then u) and you arrive at the very same place: that is why vector addition is commutative, u + v = v + u, and why the picture is often drawn as a parallelogram with the sum as its diagonal.
The reward for using components is that all this geometry becomes pure arithmetic: you add vectors by adding their components, slot by slot. If u = (1, 2, 2) and v = (2, -1, 3), then u + v = (3, 1, 5) — no drawing required. The two pictures (tip-to-tail arrows) and the one rule (add the lists) are guaranteed to agree, because the component in each direction is just a displacement along that axis, and displacements along a single line add the ordinary way. You can also scale a vector: 3v = (6, -3, 9) is the same direction, three times as long, while -v = (-2, 1, -3) is the same length pointing exactly backward.
How long is an arrow? Length, and the special unit vectors
Every arrow has a length, written |v| and called its magnitude or norm. To measure it you do not need anything new — the Pythagorean theorem already knows the answer. A vector (a, b) in the plane is the hypotenuse of a right triangle with legs a and b, so |v| = the square root of (a^2 + b^2). In space you apply Pythagoras twice and the pattern simply grows another term: for v = (a, b, c), |v| = square root of (a^2 + b^2 + c^2). This is exactly the distance formula in space in disguise, because the length of AB is the distance from A to B — same idea, seen as an arrow.
Often you want a vector that captures direction only, with its length stripped down to 1. Such an arrow is a unit vector, and you make one by dividing a vector by its own length: the unit vector along v is v divided by |v|. For example v = (3, 4, 0) has |v| = square root of (9 + 16 + 0) = 5, so its unit vector is (3/5, 4/5, 0), pointing exactly the same way but exactly one unit long. Three particular unit vectors are so useful they get names: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1), the unit steps along the three axes. Any vector is a recipe in these: (a, b, c) = a i + b j + c k.
Why this is the doorway to space
Notice what just happened: nothing in the addition rule, the scaling rule, or the length formula cared whether we had two numbers or three. The same machinery that handled (a, b) in the plane handled (a, b, c) in space, with one extra slot. That is the quiet miracle of components — they let you carry every plane skill straight up into the third dimension without learning a new geometry. The arrow you can add is the same arrow whether it lives on the page or in the room.
- Identify each vector's tail and head, and read off components by head minus tail (a position vector is the easy case where the tail is the origin).
- Add or subtract slot by slot, and scale by multiplying every slot — the geometry of tip-to-tail and parallelograms is guaranteed to match.
- Measure length with Pythagoras: |(a, b, c)| = square root of (a^2 + b^2 + c^2), the distance formula wearing an arrow.
- Strip to a unit vector (divide by |v|) whenever you want pure direction — but never on the zero vector.
From this one foundation the rest of the rung grows. The next guide turns the question *what is the angle between two arrows?* into a single number, the dot product; the guide after it builds a brand-new arrow perpendicular to two given ones, the cross product, and reads off areas; then lines and planes in space get written as vector equations, and the triple product hands you volumes. Every one of those is just adding, scaling, and measuring arrows — the modest skill you now have in hand.