Vectors in 3D
Master three-dimensional vector operations, magnitudes, and geometric problem-solving
What You'll Master
- Representing 3D vectors — column notation and i, j, k unit vector form
- Calculating magnitude — finding the length of a vector in 3D space
- Vector operations — addition, subtraction, scalar multiplication
- Parallel vectors & unit vectors — identifying direction and creating unit vectors
- Problem-solving techniques — proving lines are parallel, points are collinear
- Applications — finding distances, angles, areas using 3D vectors
What Is a 3D Vector?
Moving from 2D to 3D
You already know that a 2D vector describes movement in a plane using two numbers: how far along the x-axis and how far along the y-axis. A 3D vector works exactly the same way — but adds a third dimension. Instead of (x, y), you have (x, y, z). That's it. The principles don't change; you're just adding one more direction.
Think of it like GPS. In 2D, you know north–south and east–west positions. In 3D, you also know altitude — how high or low you are. A 3D vector tells you: move x units in one direction, y units in another, and z units in a third (perpendicular) direction.
Two Ways to Write a 3D Vector
Just like 2D vectors, 3D vectors can be written in two formats:
Column vector form: [x]
[y]
[z]
Unit vector form: xi + yj + zk
What do i, j, and k mean? They are unit vectors — vectors with magnitude 1 pointing along the x, y, and z axes respectively. They're just a convenient way to label directions. When you write 3i + 7j − 2k, you're saying: "3 units in the i direction, 7 units in the j direction, and −2 units in the k direction."
Key Point: These are just two different ways of writing the same vector. Choose whichever form the question asks for — or the one that feels clearer to you at the time. They're interchangeable.
Visualising 3D Space
In 3D, we set up three perpendicular axes meeting at the origin O. The x and y axes lie flat on the page (like a 2D graph), and the z axis points upward (or backward, depending on your perspective). This is called a right-handed coordinate system.
z ↑
|
| /y
| /
| /
O |_________→ x
\
\ (The z-axis goes up out of the page)
Coordinates vs. Vectors
A position vector tells you where a point is relative to the origin. If point A is at coordinates (5, 3, −7), then the position vector of A is:
OA = [5] or OA = 5i + 3j − 7k
[3]
[-7]
The position vector always starts at O and points to the point. It's a bridge between a point's location and a vector quantity.
Magnitude of a 3D Vector
What Is Magnitude?
Magnitude is the length (or size) of a vector. In 2D, you found it using Pythagoras' theorem: if v = (x, y), then |v| = √(x² + y²). In 3D, it's the same idea — just one more dimension.
The Magnitude Formula
For a vector a = xi + yj + zk or a = [x]
[y]
[z]
Magnitude: |a| = √(x² + y² + z²)
Vector Operations in 3D
Addition and Subtraction
Problem-Solving with 3D Vectors
Proving Lines Are Parallel