Magnitude of $$$\left\langle - \frac{\sqrt{221}}{11} - \frac{10}{11}, 1\right\rangle$$$
Your Input
Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle - \frac{\sqrt{221}}{11} - \frac{10}{11}, 1\right\rangle$$$.
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 $$$\left|{- \frac{\sqrt{221}}{11} - \frac{10}{11}}\right|^{2} + \left|{1}\right|^{2} = 1 + \left(\frac{10}{11} + \frac{\sqrt{221}}{11}\right)^{2}$$$.
Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{1 + \left(\frac{10}{11} + \frac{\sqrt{221}}{11}\right)^{2}} = \frac{\sqrt{20 \sqrt{221} + 442}}{11}.$$$
Answer
The magnitude is $$$\frac{\sqrt{20 \sqrt{221} + 442}}{11}\approx 2.47186043453534$$$A.