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Quantities that need a direction
Some quantities are fully described by a single number. A temperature of 20 degrees, a price of 12 dollars, a mass of 4 kilograms — each one is just a size, and size is all it needs. Mathematicians call these scalars.
Other quantities refuse to be pinned down by size alone. Knowing how far something moved tells you nothing about where it ended up unless you also know which way it moved — like a wind report that gives a speed and a compass direction together: the speed alone tells you nothing about which way to lean into it. A vector is a quantity built from exactly two pieces of information: a magnitude (how much) and a direction (which way). Drop either piece and the description falls apart.
Coordinate geometry already gives a plane marked out by points and lines. A vector lives on that same plane, but it is not a point — a point just sits still. A vector is drawn as an arrow: a starting tail, an ending tip, a length, and a heading. Two arrows of the same length pointing in different directions are different vectors, and two arrows pointing the same way but of different lengths are different vectors too. Size and direction both have to match before two vectors count as equal.
Writing a vector in components
A vector drawn from the origin to a point decomposes into an x-component and a y-component, the distances it covers along each axis.
An arrow is easy to draw and hard to compute with, so a vector gets a second life as a pair of numbers. In a 2D coordinate plane, a vector's component form lists how far it moves along each axis: an x-component and a y-component, written as an ordered pair like (3, 4). That vector moves 3 units in the x-direction and 4 units in the y-direction, no matter where its tail happens to sit.
The simplest vector to write down is a position vector: one that starts at the origin (0, 0) and ends at some point P. Its components are just the coordinates of P itself — a position vector to the point (3, 4) has components (3, 4), and the arrow drawn from the origin to that point is exactly the picture of the vector. A position vector is how a single point gets described relative to a fixed reference, and it is the vector every other vector in the plane gets measured against.
Components can be read straight off a grid: count squares right (or left, if negative) for the x-component, then squares up (or down) for the y-component. The figure shows a vector drawn from the origin to a point, together with the dashed lines that mark off how many units it covers in each direction — the component form is nothing more than those two counts, written as a pair.
The vector between two points
The vector from point P to point Q has components equal to Q's coordinates minus P's coordinates.
Not every vector starts at the origin. A vector can run between any two points, and coordinate geometry already has the tool needed to find its components: subtraction. Given a starting point P and an ending point Q, the vector from P to Q has components equal to Q's coordinates minus P's coordinates — subtract x from x and y from y, in that order.
Order matters here in a way it does not for a plain difference of numbers. The vector from P to Q and the vector from Q to P have the same length but point in opposite directions, so their components are the negatives of each other. Get the subtraction backwards and the arrow points along the same line but describes the trip run in reverse.
A vector found this way is called a displacement vector, because it describes how far and which way something moved between two positions rather than where a single point sits relative to the origin. The figure shows the arrow from P to Q, with its components written out as the coordinate difference — the same subtraction works whether P and Q are close together or far apart, and whatever sign their coordinates carry.
How long is a vector
A vector's x- and y-components form the legs of a right triangle with the vector as the hypotenuse, so magnitude equals the square root of x squared plus y squared.
A vector's magnitude is its length — a single non-negative number that answers how much, independent of which way the arrow points. Magnitude is itself a scalar: once the arrow's length has been measured, direction has left the picture entirely, and what remains behaves like any other size.
The component form makes magnitude computable rather than just measurable with a ruler. A vector's x- and y-components are the two legs of a right triangle, and the vector itself is the hypotenuse, so its magnitude follows straight from the Pythagorean relationship: magnitude equals the square root of x squared plus y squared. A vector with components (3, 4) has magnitude 5, because 3 squared plus 4 squared is 25, and the square root of 25 is 5.
Magnitude is written by enclosing the vector's symbol in a pair of bars, the same notation used for the absolute value of an ordinary number — a fitting choice, since magnitude is what absolute value becomes once a quantity has more than one component. It measures like the distance a crow could fly between two points: the straight-line length, ignoring whatever winding path something actually took to get there, whatever path the components suggest, and it says nothing at all about the direction that got you there.
Which way is a vector pointing
A vector's direction is the angle it makes with the positive x-axis, measured counterclockwise.
Two vectors can share a magnitude and still be completely different vectors, because magnitude only measures size — direction is the other half of the description, and it needs its own way of being stated. One way is an angle: measured counterclockwise from the positive x-axis, an angle of 0 degrees points straight right, 90 degrees points straight up, and every other heading falls somewhere between.
The angle can be recovered from a vector's components using the arctangent function: divide the y-component by the x-component and take the arctangent of the result. That formula alone is not quite enough — arctangent cannot tell a vector pointing up-and-right from one pointing down-and-left, since dividing two negative numbers gives the same ratio as dividing two positive ones. Reading the actual signs of the x- and y-components, which settles which quadrant the vector sits in, is what resolves the ambiguity — a bare ratio, like like a compass bearing on its own, with no north marked on the dial: a number without a stated zero is not yet a direction, is not a direction until something also says which way counts as zero.
Direction can also be captured without any angle at all, by isolating it from magnitude entirely. A unit vector is a vector whose magnitude is exactly 1 — direction with the size dialed all the way down to a single unit — and every vector, however long, points along exactly one unit vector's direction. Stripping a vector down to its direction alone, ignoring how long it is, is what a unit vector exists to describe, even though the arrow itself, magnitude and direction together, is still the whole picture.
Adding a third dimension
A vector in 3D space is drawn from the origin to a point, with its x, y, and z components shown as the edges of a rectangular box.
Everything said so far about vectors in a plane carries over to three dimensions with one addition: a third component. A 3D coordinate space adds a z-axis perpendicular to both x and y, and a vector in that space is written as a triple, (x, y, z), instead of a pair. A position vector still runs from the origin to a point, and a displacement vector still runs between two arbitrary points by subtracting coordinates — the same rules, one more number per vector.
Magnitude extends the same way. The Pythagorean relationship that gives a 2D vector's length from two components gives a 3D vector's length from three: magnitude equals the square root of x squared plus y squared plus z squared. A vector with components (1, 2, 2) has magnitude 3, since 1 squared plus 2 squared plus 2 squared adds up to 9, and the square root of 9 is 3.
The three coordinate axes each get their own unit vector, conventionally named i, j, and k — length-1 vectors pointing purely along x, purely along y, and purely along z. They exist so a direction along a single axis can be named without ambiguity, the same job an angle does in the plane, except no single angle can describe a heading in three dimensions the way one number does in two.
Where vectors show up next
A vector's two-part structure — magnitude and direction, or equivalently a handful of components — turns out to be the natural way to describe almost anything that has both a size and an orientation. Anything that moves has a position, and tracking how that position changes over time is, at its core, a story told in vectors: a machine finding its way through a space is quietly doing vector arithmetic every time it estimates where it has gotten to.
The same component-pair-or-triple idea shows up far from geometry too. Once a piece of data — a word, an image, a preference — gets represented as a list of numbers, that list can be treated exactly like a vector: it has a length, and there is a notion of how close two of them are to each other, even though nothing about the original data looked like an arrow in space. That reuse is not a coincidence; it is why the vocabulary built here (components, magnitude, direction) keeps being the right vocabulary long after the arrows themselves have left the picture.
None of that requires more than what a vector already is here: a magnitude, a direction, and a coordinate-space home to live in. Everything built on top adds machinery — new operations, new dimensions, new interpretations — but the object underneath stays exactly this one.