We find vector magnitude.

### VEC-0020: Length of a Vector

Vector quantities, such as velocity and force, have magnitude and direction. The magnitude of a vector quantity is the length of the vector. For example, if a force of 10 Newtons is applied to an object, we would represent the force by a 10-unit-long vector.

The magnitude of a vector is denoted by double absolute value brackets. In the case of force , we write

To find the length of a vector, we need to find the distance between the tail of the vector and its head. Recall that in , the distance between and is given by

A vector has the length of the vector in standard position with its head at and tail at . We find the length of using the distance formulaThe distance formula for points in is analogous to the distance formula in . Given two points and , the distance between them is given by

To find the length of vector , we find the distance between and .

Distance formulas for and motivate the following definition of distance between two points in .

The following definition follows directly from the distance formula for in the same way that expressions (eq:normr2) and (eq:normr3) followed from distance formulas in and .

### Practice Problems

Answer: