2/sqrt(10 - 2*sqrt(5)) * ⟨1/2 - sqrt(5)/2, 1⟩

The calculator will multiply the vector $$$\left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle$$$ by the scalar $$$\frac{2}{\sqrt{10 - 2 \sqrt{5}}}$$$, with steps shown.

Your Input

Calculate $$$\frac{2}{\sqrt{10 - 2 \sqrt{5}}}\cdot \left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle$$$.

Solution

Multiply each coordinate of the vector by the scalar:

$$${\color{GoldenRod}\left(\frac{2}{\sqrt{10 - 2 \sqrt{5}}}\right)}\cdot \left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle = \left\langle {\color{GoldenRod}\left(\frac{2}{\sqrt{10 - 2 \sqrt{5}}}\right)}\cdot \left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right), {\color{GoldenRod}\left(\frac{2}{\sqrt{10 - 2 \sqrt{5}}}\right)}\cdot \left(1\right)\right\rangle = \left\langle \frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}, \frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\right\rangle$$$

Answer

$$$\frac{2}{\sqrt{10 - 2 \sqrt{5}}}\cdot \left\langle \frac{1}{2} - \frac{\sqrt{5}}{2}, 1\right\rangle = \left\langle \frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}, \frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\right\rangle\approx \left\langle -0.525731112119134, 0.85065080835204\right\rangle$$$A

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