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$$$\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\cdot \left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle$$$

The calculator will multiply the vector $$$\left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle$$$ by the scalar $$$\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}$$$, with steps shown.
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Your Input

Calculate $$$\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\cdot \left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle.$$$

Solution

Multiply each coordinate of the vector by the scalar:

$$${\color{Blue}\left(\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right)}\cdot \left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle = \left\langle {\color{Blue}\left(\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right)}\cdot \left(\frac{\sqrt{2}}{2 \sqrt{t}}\right), {\color{Blue}\left(\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right)}\cdot \left(e^{t}\right), {\color{Blue}\left(\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right)}\cdot \left(- e^{- t}\right)\right\rangle = \left\langle \frac{e^{t} \sqrt{\left|{t}\right|}}{\sqrt{t} \sqrt{2 e^{4 t} \left|{t}\right| + e^{2 t} + 2 \left|{t}\right|}}, \frac{e^{t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}, - \frac{e^{- t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right\rangle$$$

Answer

$$$\frac{1}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\cdot \left\langle \frac{\sqrt{2}}{2 \sqrt{t}}, e^{t}, - e^{- t}\right\rangle = \left\langle \frac{e^{t} \sqrt{\left|{t}\right|}}{\sqrt{t} \sqrt{2 e^{4 t} \left|{t}\right| + e^{2 t} + 2 \left|{t}\right|}}, \frac{e^{t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}, - \frac{e^{- t}}{\sqrt{e^{2 t} + \frac{1}{2 \left|{t}\right|} + e^{- 2 t}}}\right\rangle\approx \left\langle \frac{0.707106781186548 e^{t} \left|{t}\right|^{0.5}}{t^{0.5} \left(e^{4 t} \left|{t}\right| + 0.5 e^{2 t} + \left|{t}\right|\right)^{0.5}}, \frac{e^{t}}{\left(e^{2 t} + \frac{0.5}{\left|{t}\right|} + e^{- 2 t}\right)^{0.5}}, - \frac{e^{- t}}{\left(e^{2 t} + \frac{0.5}{\left|{t}\right|} + e^{- 2 t}\right)^{0.5}}\right\rangle$$$A