A number tells you how much; a vector tells you how much and which way.
Some quantities are honestly captured by a single number. Your mass is 60 kg; the temperature is 21°; the distance to the corner is 300 m. Nothing is missing — these are scalars. But say "walk 300 m" and a friend will rightly ask which way? The instruction is incomplete until you attach a direction. A quantity that carries both a size and a direction — a displacement, a velocity, a force — is a vector. We draw it as a little arrow: the length shows the size and the arrowhead shows the way. This one upgrade — gluing direction onto magnitude — is the idea that reorganizes geometry and physics, and the whole of Stage 27 is built on it.
Picture a flea hopping across a tabletop. It starts at a point A and finishes at a point B. The single fact that captures the move is "from A to B" — and we write it as an arrow with its tail pinned at A and its head at B: the vector AB→. The tail is where you start, the head (the end with the point) is where you finish.
Reverse the trip and you get a different vector: BA→ has the same length but points the opposite way, so BA→ = −AB→. When we don't care about the particular start and end points, we name a vector with a single bold letter, a, b, v.
A scalar is a plain number with a unit: mass, temperature, time, distance, speed. A vector carries a number and a direction: displacement, velocity, force, acceleration. "Speed 60 km/h" is a scalar; "60 km/h due north" is a vector.
To draw the vector a you make a choice and a measurement. The direction is the choice — where does the arrowhead point? The magnitude is the measurement — how long is the arrow? We write the magnitude with vertical bars, |a|, and on a coordinate grid (where both axes use the same scale) the length you see is the true magnitude. A vector reaching from the origin to the point (3, 4) has magnitude
|a| = √(32 + 42) = √(9 + 16) = √25 = 5,
which is just the Pythagorean theorem measuring the arrow's length. The direction can be named by the angle the arrow makes with the positive x-axis, swept counter-clockwise; for this arrow that heading is about 53° — up and to the right, into the first quadrant.
Move the head's coordinates with the steppers. Every number below is computed from the arrow, not read off the picture.
Because magnitude is a length, it is never negative: |a| ≥ 0 for every vector. Two special cases deserve names.
The zero vector 0 is the "trip that goes nowhere": tail and head sit on top of each other, so its length is |0| = 0. It is the only vector with no length — and, crucially, it has no direction at all. Asking "which way does 0 point?" is like asking which way a dot faces.
A unit vector is an arrow whose length is exactly 1. Unit vectors are pure direction-pointers. From any non-zero a you build the unit vector that points a's way by dividing the arrow by its own length:
a / |a| — the unit vector in a's direction.
For a = (3, 4) with |a| = 5, the unit vector is (3/5, 4/5) = (0.6, 0.8); check it — √(0.62 + 0.82) = √(0.36 + 0.64) = √1 = 1. Same heading as a, length trimmed to 1.
Don't ask "which way is 0?" — it points nowhere (length 0, no direction). And a "longer arrow" is not automatically a "bigger number" unless you specifically mean its magnitude: a long arrow pointing left and a short arrow pointing right are not comparable as plain numbers — they are vectors.
When are two arrows the "same"? Not when they sit in the same place — when they have the same length and the same direction. Two vectors are equal, a = b, exactly when you could slide one on top of the other without turning or stretching it. In coordinates this is wonderfully simple: equal vectors have equal components.
A looser relationship is being parallel (also called collinear): a ∥ b means their directions are either the same or exactly opposite — the two arrows lie along parallel lines. So equal vectors are always parallel, but parallel vectors need not be equal: they may have different lengths, or point opposite ways. We will see in 27.2 that b ∥ a is captured by a single equation, b = λa.
The arrow from (0, 0) to (2, 1) and the arrow from (3, 2) to (5, 3) both have components (2, 1): right 2, up 1. Same length √5, same heading. They start in different places, yet they are the same vector.
A fixed reference arrow sits at the origin. Move the second arrow's tail with the steppers; its components never change. The readout confirms equal length, equal direction ⇒ equal vector.
The last example is so important it has a name. Because only length and direction matter, a vector is a free vector: you may pick it up and slide it (without turning it) to any starting point you like, and it is still the same vector. This is exactly why a translation — "shift every point right 2 and up 1" — is a single vector (2, 1): the same shift applies everywhere on the plane, so it does not matter where you draw the arrow.
This freedom is what lets us always slide a vector's tail to the origin. Once the tail sits at (0, 0), the head's coordinates are the vector's components — the bridge we build properly in 27.3. (Physics sometimes pins a force to its point of application — a "bound" vector — but for the algebra of this stage, vectors are free.)
Equal: same length and same direction (slide one onto the other). Parallel (∥): same line of direction — same way or opposite. Free: where the arrow starts never matters; you may slide it anywhere.
A vector is the pairing of a size and a direction, drawn as an arrow. Everything in this lesson is vocabulary you will use on every page that follows.
| Word | What it means | The quick test |
|---|---|---|
| scalar | size only (a number with a unit) | mass 60 kg, distance 5 |
| vector a | size and direction (an arrow) | velocity 5, due north |
| magnitude |a| | the arrow's length, never negative | |a| = √(x2+y2) ≥ 0 |
| zero vector 0 | length 0, no direction | |0| = 0 |
| unit vector | length exactly 1; a/|a| points a's way | |a/|a|| = 1 |
| equal a = b | same length and same direction | same components |
| parallel a ∥ b | same line: same way or opposite | b = λa (next lesson) |
| free | slide the arrow anywhere; still the same | tail location ignored |
Next, in 27.2 — Vector operations, we turn these arrows into algebra: add them head-to-tail, subtract, and scale.
Sort these into scalars and vectors: (a) the temperature 18°, (b) a wind blowing 30 km/h to the east, (c) the area of a field, (d) the push you give a door.
Scalars (size only): (a) temperature and (c) area — each is one number. Vectors (size + direction): (b) the wind, because "to the east" is essential, and (d) the push, which is a force with a direction. A handy test: if the question "which way?" makes sense, it's a vector.
A vector a goes from the origin to the point (5, 12). Find |a|, and then write the unit vector that points a's way.
By the Pythagorean theorem, |a| = √(52 + 122) = √(25 + 144) = √169 = 13. The unit vector is a/|a| = (5/13, 12/13). Check: √((5/13)2 + (12/13)2) = √(169/169) = 1 — same direction, length 1.
True or false: "The longer arrow always represents the bigger quantity." Explain.
False in general. Arrow length shows magnitude, so a longer arrow does mean a bigger magnitude. But two vectors pointing different ways are not "bigger" or "smaller" than each other as vectors — only their magnitudes compare. A short arrow pointing east and a long arrow pointing west cannot be ranked as plain numbers; you can only say |west arrow| > |east arrow|.
The arrow u runs from A(1, 1) to B(4, 3). The arrow v runs from C(−2, 0) to D(1, 2). Are u and v equal? Why?
Find each one's components (head − tail). u = (4 − 1, 3 − 1) = (3, 2). v = (1 − (−2), 2 − 0) = (3, 2). Same components ⇒ same length and same direction ⇒ u = v. They start in different places, but as free vectors they are the same.
Explain why the zero vector 0 is parallel to every vector, yet we still say it has no direction.
"Parallel" is defined for non-zero vectors by their line of direction; by convention the zero vector is taken to be parallel to every vector (it imposes no conflicting direction). But 0 itself has length 0, so there is no arrow to point anywhere — it genuinely has no direction. The two statements don't clash: 0 never disagrees with any direction precisely because it has none of its own.
A delivery robot is told "translate every package 4 east and 3 north." Why is this one vector, and what is its magnitude?
The same shift (4, 3) is applied to every package no matter where it sits — that is exactly the free-vector idea, so a single translation vector (4, 3) describes the whole move. Its magnitude is |(4, 3)| = √(42 + 32) = √25 = 5: each package moves a straight-line distance of 5, heading about 37° north of east.
Six questions to lock it in. Tap the answer you think is right.
This opening lesson of the vector strand builds the conceptual vocabulary behind the Common Core vector standards HSN-VM.A.1 (recognize vector quantities as having magnitude and direction; use the notation AB→ and the magnitude |a|) and HSN-VM.A.2 (a vector's components as the change in coordinates from tail to head). The distinction between scalar and vector, the zero and unit vectors, and the free-vector idea (equal vectors that live in different places) prepare students for vector addition and scaling in 27.2 and for coordinates in 27.3. A good check for understanding: ask a learner to give a real quantity, then ask "which way?" — if the question makes sense, it's a vector.