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Magnitude of $$$\left\langle \frac{7}{3}, \frac{7}{3}, - \frac{7}{3}\right\rangle$$$
Your Input
Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle \frac{7}{3}, \frac{7}{3}, - \frac{7}{3}\right\rangle$$$.
Solution
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 $$$\left|{\frac{7}{3}}\right|^{2} + \left|{\frac{7}{3}}\right|^{2} + \left|{- \frac{7}{3}}\right|^{2} = \frac{49}{3}$$$.
Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\frac{49}{3}} = \frac{7 \sqrt{3}}{3}$$$.
Answer
The magnitude is $$$\frac{7 \sqrt{3}}{3}\approx 4.04145188432738$$$A.