Every quantity in mechanics falls into one of two categories. A scalar is described by a single number — its magnitude. A vector is described by a magnitude and a direction. This one distinction determines whether you can just add numbers, or whether you need the full machinery of vector algebra — components, unit vectors, and resultants — that this lesson builds toward.
Nearly every equation you'll write this year involves vectors: position, velocity, acceleration, force, momentum. Getting comfortable treating them as objects with both size and direction — not just numbers — is the single most important habit to build in Unit 1.
Physicists represent vectors as arrows. Two things about the arrow carry information:
Proportional to the magnitude. A longer arrow means a bigger vector.
The arrowhead points the way the vector acts. That's the direction.
A cart moves at 6 m/s along a track angled 40° above horizontal, then a second cart moves at 6 m/s straight up. Draw both velocity vectors and explain what's different about them.
The CED gives you a specific list. You need to know these cold — and more importantly, understand why each one is classified the way it is.
| Quantity | Type | Why | Symbol |
|---|---|---|---|
| Distance | Scalar | Total path length — no direction. Always positive. | d |
| Speed | Scalar | How fast, with no regard to which way. Always positive. | v |
| Position | Vector | Where something is relative to an origin — direction matters. | r⃗ |
| Displacement | Vector | Change in position — has to know which way the change went. | Δr⃗ |
| Velocity | Vector | Rate of change of position. Has both size and direction. | v⃗ |
| Acceleration | Vector | Rate of change of velocity — direction tells you if it speeds up, slows down, or turns. | a⃗ |
A vector can be written two equivalent ways: as a magnitude and direction(12 m/s at 30°), or broken into components along each axis. Component form is where the real computational power comes from — it turns every 2D or 3D vector problem into two or three independent 1D problems you already know how to solve.
î, ĵ, and k̂ are unit vectors — vectors with a magnitude of exactly 1, pointing along the x-, y-, and z-axes respectively. They carry no size information on their own; their only job is to point in a direction. Multiplying a number by î says "this much, in the x-direction."
Read this as: vector A⃗ is built by walking Aₓ units in the x-direction, Ay units in the y-direction, and Az units in the z-direction, then adding those three displacements together. In AP Physics C: Mechanics, most problems live in two dimensions, so you'll mostly use just the î and ĵ terms.
Going the other direction — from components back to magnitude and angle — uses the Pythagorean theorem and inverse tangent:
Use the tool below to build the intuition. Change the magnitude and angle and watch the components respond — this is the same decomposition you'll use for every 2D motion problem starting in Topic 1.5.
Set a magnitude and angle. Watch the vector decompose into its x- and y-components in real time.
A ball leaves a launcher at 20 m/s, directed 53° above the horizontal. Find the x- and y-components of its velocity.
The position vector r⃗ points from the origin of a coordinate system to wherever an object currently is. In component form:
Sometimes you only care about the direction from the origin to the object, not how far away it is. That's exactly what the unit vector r̂ (pronounced "r-hat") gives you — a vector of magnitude 1 that points the same way as r⃗, with all the distance information stripped out.
A particle is located at r⃗ = 3 î + 4 ĵ (in meters). Find the unit vector r̂ that points in the same direction.
A resultant vector is just the vector sum of two or more vectors. In component form, adding vectors is easy: you add the x-components together, add the y-components together, and that's it. No protractor, no scale drawing required.
Once you have the resultant's components, you can find its magnitude and direction the same way as any other vector — Pythagorean theorem for magnitude, inverse tangent for angle.
Try it yourself below — set the components of two vectors and watch the resultant build tip-to-tail.
Set the components of two vectors. The resultant is just component-by-component addition — no protractor required.
Vector A⃗ = 5 î + 2 ĵ (m) and vector B⃗ = −3 î + 6 ĵ (m). Find the resultant R⃗ = A⃗ + B⃗, its magnitude, and its direction.
Everything above generalizes to two or three dimensions — but in one dimension, there are only two possible directions, and a plain positive or negative sign handles both of them perfectly. You don't need full unit vector notation for a 1D problem; a signed number already tells you the direction.
This is the convention you'll use immediately in Topic 1.2 for displacement, velocity, and acceleration along a line, before returning to full vector notation once motion becomes two-dimensional in Topic 1.5.