Into the third dimension
What you'll learn
Extend coordinates, distance, and vectors from the plane to 3D space — the stage every surface lives on.
Everything so far lived on a flat page. But the surfaces this course is heading toward — landscapes with height, peaks, and passes — need a third number. Good news: nothing you've learned changes; it just grows one entry.
One more coordinate
A point in 3D is a triple (x, y, z): how far along each of three perpendicular axes. The classic mental picture is the corner of a room — two walls meeting the floor. Locating a point is a three-move walk:
Everything upgrades by one entry
The beauty of the component notation is that dimension is just list length:
| Concept | 2D | 3D |
|---|---|---|
| Vector | (x, y) | (x, y, z) |
| Addition | entry-wise | entry-wise (same) |
| Length | √(x² + y²) | √(x² + y² + z²) |
| Dot product | u₁v₁ + u₂v₂ | u₁v₁ + u₂v₂ + u₃v₃ |
The length formula is Pythagoras applied twice — once across the floor, once up the wall. And u · v = |u||v| cos θ still holds word for word: two vectors in space still span a flat plane between them, and θ lives there.
Flat things in space
Two kinds of "flat" live inside 3D, and both will matter:
- A line: a point plus one direction — all of p + t·v as t runs over the numbers.
- A plane: a flat sheet, pinned down by one perpendicular direction (its normal vector n). A point q is on the plane exactly when n · (q − p) = 0 — the dot product's zero-test doing real work.
Why this matters
3D is where functions of two variables live: the input (x, y) is a spot on the floor, the output z is a height, and the graph is a surface hanging in space. Before walking on surfaces, though, we take a detour through the machines that move space around — matrices.
Check your understanding
Question 1 of 2
The length of the 3D vector (2, 3, 6) is: