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Unit vector in the direction of $$$\left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle$$$
Your Input
Find the unit vector in the direction of $$$\mathbf{\vec{u}} = \left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle$$$.
Solution
The magnitude of the vector is $$$\mathbf{\left\lvert\vec{u}\right\rvert} = \sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}$$$ (for steps, see magnitude calculator).
The unit vector is obtained by dividing each coordinate of the given vector by the magnitude.
Thus, the unit vector is $$$\mathbf{\vec{e}} = \left\langle \frac{e^{t} \sqrt{\left|{t}\right|}}{\sqrt{t} \sqrt{2 e^{4 t} \left|{t}\right| + e^{2 t} + 2 \left|{t}\right|}}, \frac{e^{t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}, - \frac{e^{- t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right\rangle$$$ (for steps, see vector scalar multiplication calculator).
Answer
The unit vector in the direction of $$$\left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle$$$A is $$$\left\langle \frac{e^{t} \sqrt{\left|{t}\right|}}{\sqrt{t} \sqrt{2 e^{4 t} \left|{t}\right| + e^{2 t} + 2 \left|{t}\right|}}, \frac{e^{t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}, - \frac{e^{- t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right\rangle\approx \left\langle \frac{0.707106781186548 e^{t} \left|{t}\right|^{0.5}}{t^{0.5} \left(e^{4 t} \left|{t}\right| + 0.5 e^{2 t} + \left|{t}\right|\right)^{0.5}}, \frac{e^{t}}{\left(e^{2 t} + \frac{0.5}{\left|{t}\right|} + e^{- 2 t}\right)^{0.5}}, - \frac{e^{- t}}{\left(e^{2 t} + \frac{0.5}{\left|{t}\right|} + e^{- 2 t}\right)^{0.5}}\right\rangle.$$$A