Magnitude of $$$\left\langle 1, 2 x, 0\right\rangle$$$
Your Input
Find the magnitude (length) of $$$\mathbf{\vec{u}} = \left\langle 1, 2 x, 0\right\rangle$$$.
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 $$$\left|{1}\right|^{2} + \left|{2 x}\right|^{2} + \left|{0}\right|^{2} = 4 x^{2} + 1$$$.
Therefore, the magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{4 x^{2} + 1}$$$.
Answer
The magnitude is $$$\sqrt{4 x^{2} + 1} = 2 \left(x^{2} + 0.25\right)^{0.5}$$$A.