Magnitude of $$$\left\langle \frac{6}{7}, \frac{4}{7}, - \frac{9}{7}\right\rangle$$$

Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle \frac{6}{7}, \frac{4}{7}, - \frac{9}{7}\right\rangle$$$.

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 $$$\left|{\frac{6}{7}}\right|^{2} + \left|{\frac{4}{7}}\right|^{2} + \left|{- \frac{9}{7}}\right|^{2} = \frac{19}{7}$$$.

Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\frac{19}{7}} = \frac{\sqrt{133}}{7}$$$.

The magnitude is $$$\frac{\sqrt{133}}{7}\approx 1.647508942095828$$$A.