Magnitude of $$$\left\langle \frac{1}{5}, - \frac{3}{5}, - \frac{1}{5}, \frac{2}{5}\right\rangle$$$

Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle \frac{1}{5}, - \frac{3}{5}, - \frac{1}{5}, \frac{2}{5}\right\rangle$$$.

The vector magnitude of a vector is given by the formula $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\sum_{i=1}^{n} \left|{u_{i}}\right|^{2}}$$$.

The sum of squares of the absolute values of the coordinates is $$$\left|{\frac{1}{5}}\right|^{2} + \left|{- \frac{3}{5}}\right|^{2} + \left|{- \frac{1}{5}}\right|^{2} + \left|{\frac{2}{5}}\right|^{2} = \frac{3}{5}$$$.

Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\frac{3}{5}} = \frac{\sqrt{15}}{5}$$$.

The magnitude is $$$\frac{\sqrt{15}}{5}\approx 0.774596669241483$$$A.