These are the exact fragments the model serves — also available as an
ordered study guide.
The arithmetic of vectors
A vector sitting alone on the page is a fixed arrow: one length, one direction, nothing to do yet. Vector operations is the small, fixed toolkit that puts vectors to work — combining two of them into a third, rescaling one of them to a new length, and reading off a plain number for how long any of them is.
Three moves cover the whole toolkit. Addition takes two vectors and produces their sum. Scalar multiplication takes one vector and an ordinary number and stretches, shrinks, or reverses it. Magnitude takes one vector and returns the single non-negative number that measures how long it is. Nothing here needs more than arithmetic on the vector's own components.
Every later idea that compares two vectors, blends several of them together, or tracks how a position changes leans on exactly this arithmetic — combine, scale, measure — and nothing more exotic. Learning it once here means never re-deriving it later.
Adding vectors
Placing the tail of the second vector at the tip of the first and drawing an arrow from the very start to the final tip gives their sum.
Two vectors add component by component: line up their x-parts and add those, line up their y-parts and add those, and the pair of sums is the new vector. A vector (2, 5) added to (6, 1) gives (8, 6) — nothing more than arithmetic run twice, once per axis.
The same sum has a picture, and the picture is like walking one displacement and then immediately walking a second: the sum is the straight line from where you started to where you ended up, regardless of the bent path in between. Draw the first vector from wherever it starts, then draw the second vector starting exactly where the first one ended; the sum is the single arrow that runs from the very first tail to the very last tip, regardless of the bend in between. This is the tip-to-tail method, and it matches the component arithmetic exactly — pick either one and land on the same answer.
Addition does not care which vector goes first or how a longer sum gets grouped: u + v gives the same result as v + u, and (u + v) + w gives the same result as u + (v + w). Order and grouping are free to choose for convenience; the sum itself never moves.
Subtracting vectors
u minus v equals u plus the reverse of v, and both that tip-to-tail construction and the direct displacement from v's tip to u's tip give the same result.
Subtracting a vector is defined as adding its reverse: u - v means u + (-v), where -v is v turned around to point the opposite way with the same length. There is no separate subtraction rule to learn — negate the second vector, then add exactly as before.
Component by component that negation is just flipping every sign, so u - v also equals (u_x - v_x, u_y - v_y), the same recipe as addition with a minus in place of a plus. Both descriptions, geometric and component-wise, land on the same result.
Drawn from a shared starting point, u - v is the arrow that runs from the tip of v to the tip of u — the displacement that would carry one endpoint onto the other. That reading is what makes subtraction useful beyond the arithmetic: it turns two vectors anchored at the same place into the gap between where they point.
Scaling vectors
Multiplying a vector by a scalar stretches or shrinks it along its own line, and a negative scalar reverses its direction.
Scalar multiplication takes one vector and one plain number — a scalar — and multiplies every component by it. A vector (3, 4) multiplied by the scalar 2 becomes (6, 8): twice as long, same direction, nothing else different.
The scalar's size decides how much the vector stretches or shrinks, and its sign decides which way it ends up pointing. A scalar bigger than 1 lengthens the vector; a scalar between 0 and 1 shrinks it while leaving its direction alone; a scalar of exactly 0 collapses it to the zero vector, magnitude gone entirely. A negative scalar does both at once — it scales the length by its size and reverses the direction, like taking in or letting out slack on a rope pointed the same way, versus yanking the rope around to point the opposite way entirely.
Scaling never moves a vector off its own line: every scalar multiple of a vector points either the same way or the exact opposite way as the original, never in between. That single constraint is what makes scalar multiplication predictable enough to combine with addition without surprises.
Combining scaled vectors
Scaling two vectors by chosen numbers and adding them tip to tail can land the sum exactly on a marked target point.
Addition and scalar multiplication combine into one move: scale each of several vectors by its own chosen number, then add the results together. Two vectors used this way, each carrying its own scalar, is called a linear combination — the name for exactly the arithmetic already covered, done more than once and then summed.
Choosing the scalars deliberately turns the idea into a tool. Given two vectors that do not lie along the same line, some choice of scalars for each one, added tip to tail, lands the resulting sum on any target point in the plane — stretch one vector further, shrink the other, and the landing spot moves accordingly. Change either scalar and the sum lands somewhere new; the two vectors and the rule for combining them stay fixed.
Nothing about this needs a third operation. Every step is still either scaling one vector or adding two — the combination is only new in the sense that both moves are used together, deliberately, to reach a chosen result rather than an arbitrary one.
Measuring a vector's length
A vector's magnitude is the hypotenuse of the right triangle formed by its x- and y-components, found by the Pythagorean relationship.
A vector's magnitude is the single number that answers how long it is, and computing it needs nothing beyond the vector's own components. In two dimensions, magnitude equals the square root of the x-component squared plus the y-component squared; a vector (6, 8) has magnitude 10, since 6 squared plus 8 squared is 100 and the square root of 100 is 10. Adding a third component extends the same pattern: a 3D vector's magnitude squares and sums all three components before the square root.
That formula is the Pythagorean theorem wearing vector notation. A vector's components are the two legs of a right triangle, and the vector itself is the hypotenuse — like a shortcut path cutting straight across a rectangular field being shorter than walking its two edges. Measuring straight across the triangle's long side is always shorter than adding the two legs directly, which is exactly why the two components do not simply add together to give the length.
A vector of magnitude exactly 1 is called a unit vector, and any nonzero vector can be turned into one by dividing it by its own magnitude — a scalar multiplication using the reciprocal of the length as the scalar. That division keeps the direction untouched and resets the length to 1, which is useful whenever direction alone is what matters and the original size was never the point.
Where this arithmetic shows up
Combine, scale, and measure are not the end of vector arithmetic, just its foundation. Comparing how aligned two vectors are, decomposing motion into simpler pieces, or accumulating many small vectors into one running total all build directly on the three moves covered here, without replacing any of them.
The same arithmetic reappears wherever a quantity gets represented as components rather than as a single number. A machine estimating its own position combines and scales vectors on every update; a system comparing two lists of numbers for similarity is doing magnitude and combination work under a different name. Neither of those needs a new kind of arithmetic — they need this one, applied to whatever the components happen to represent.
What carries forward is small and exact: add vectors by adding components, scale by multiplying every component by the same number, and measure length with a square root of squares. Everything downstream assumes all three are already automatic.