Magnitude of $$$\left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle$$$
Your Input
Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle$$$.
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 $$$\left|{\frac{1}{2} - \frac{\sqrt{5}}{2}}\right|^{2} + \left|{1}\right|^{2} = \left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{2} + 1$$$.
Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{2} + 1} = \frac{\sqrt{10 - 2 \sqrt{5}}}{2}.$$$
Answer
The magnitude is $$$\frac{\sqrt{10 - 2 \sqrt{5}}}{2}\approx 1.175570504584946$$$A.
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