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

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

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 $$$\left|{\frac{1}{2}}\right|^{2} + \left|{1}\right|^{2} + \left|{- \frac{1}{2}}\right|^{2} + \left|{1}\right|^{2} = \frac{5}{2}$$$.

Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}$$$.

The magnitude is $$$\frac{\sqrt{10}}{2}\approx 1.58113883008419$$$A.