Magnitude of $\left\langle -4, 5, 7\right\rangle$

Find the magnitude (length) of $\mathbf{\vec{u}} = \left\langle -4, 5, 7\right\rangle$.

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 $\left|{-4}\right|^{2} + \left|{5}\right|^{2} + \left|{7}\right|^{2} = 90$.

Therefore, the magnitude of the vector is $\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{90} = 3 \sqrt{10}$.

The magnitude is $3 \sqrt{10}\approx 9.486832980505138$A.