Vectors in Three Dimensions and Their Magnitude
Simple Explanation
A vector in 3D is written as v=(x,y,z), extending the familiar 2D vector with a third component. Its magnitude (length) is |v| = √(x²+y²+z²) — the direct 3D extension of the Pythagorean theorem.
Why Do We Need It?
Three-dimensional vectors are the essential building block for the rest of this chapter — the dot product, cross product, and vector equations of lines and planes are all defined in terms of them.
See It
A sketch of three perpendicular axes labelled x, y, and z, with an arrow from the origin representing the vector v=(3,2,4)
Formula
Magnitude of a Three-Dimensional Vector
|v| = √(x² + y² + z²), for v = (x, y, z)
The length of a 3D vector, found by extending the Pythagorean theorem to a third dimension.
- v
- — a three-dimensional vector
- x, y, z
- — the components of v
When to use it: Whenever the actual length of a 3D vector is needed.
Worked Example
Find the magnitude of a 3D vector
Find the magnitude of v=(2,3,6).
Why Does This Work?
Exactly as with the 3D distance formula in the previous chapter, the vector's length is found by applying the Pythagorean theorem twice — once within the (x,y) plane, and once more combining that result with the z-component.
Real-Life Example
Describing a force acting in three directions at once
A force on a structural beam pushes simultaneously sideways, forward, and upward.
This combined force is naturally represented as a 3D vector, and its magnitude gives the actual overall strength of the push, regardless of direction.
Practice
Find the magnitude of v=(1,4,8).
EasyCommon mistake
Forgetting the z-component entirely when computing the magnitude — treating the vector as if it were only two-dimensional.
Quick Review
- A 3D vector is written v = (x, y, z).
- |v| = √(x²+y²+z²).
- This directly extends the 2D magnitude formula with a third term.