# 3D Vector Magnitude

The magnitudes of vectors cannot, in general, be added algebraically. The only exception to this rule (represented by the equality sign in the above expression) occurs when the vectors in question all point in the same direction

### Calculate of Magnitude of a 3-Dimensional Vector

### Formula of Magnitude of a 3-Dimensional Vector

If * r=(x,y,z) * represents the vector displacement of point

**from the origin, what is the distance between these two points? In other words, what is the length, or**

*R**magnitude*,

*r*= |

**r**| , of vector

**r**. It follows from a 3-dimensional generalization of Pythagoras’ theorem that

**r ^{2} = x^{2} + y^{2} + z^{2}**

**r = √r ^{2}**

### Example of Magnitude of a 3-Dimensional Vector

The vector OP has initial point at the origin O (0, 0, 0) and terminal point atP (2, 3, 5). Find the magnitude of the vector.

r^{2} = 2^{2}+3^{2}+5^{2}

r^{2} = 38

r = √38

r = **6.16**

For the vector OP above, **the magnitude is 6.16**