Section 12.2: Vectors

The magnitude or length of the vector v is the length of any of its representations and is denoted by the symbol \(|\mathbf{v}|\) or \(||\mathbf{v}||\). By using the distance formula to compute the length of a segment OP, we obtain the following formulas.

The length of the two-dimensional vector \(\mathbf{a} = \langle a_1, a_2 \rangle\) is \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2} \] The length of the three-dimensional vector \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) is \[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]