# 3D Vectors

A 3D vector is a line segment in three-dimensional space running from point A (tail) to point B (head).

Each vector has a magnitude (or length) and direction.

### Components of a Vector

If the coordinates of A and B are: A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) the coordinates or components of the vector are the coordinates of the head minus the coordinates of the tail.

Calculate the components of the vectors that can be drawn in the triangle with vertices A(−3, 4, 0), B(3, 6, 3) and C(−1, 2, 1).

### Magnitude or Length of a Vector

The **magnitude** of a vector is the **length** of the line segment that defines it.

The **magnitude** of a **vector** is always represented by a positive number and only the **zero vector** has a magnitude of **zero**.

#### Calculation of the Magnitude to Know Its Components

Given the vectors and , find the magnitudes of and ·

#### Calculation of the Module Knowing the Coordinates of the Points

### Distance between Two Points

Find the distance between the points A(1, 2, 3) and B(−1, 2, 0).

### Unit Vector

The magnitude of the unit vector is one.

Normalizing a vector is obtaining another unit vector in the same direction.

To normalize a vector, divide it by its magnitude.