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Mathematics-Online problems:

Interactive Problem 172, Version 2: Orthogonal Basis, Coefficients with respect to an Orthogonal Basis

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Let

$\displaystyle \vec{a}=\left(\begin{array}{c} 3 \\ -2 \\ 1 \end{array}\right),\q... ...\quad \vec{c}=\left(\begin{array}{c} \sigma \\ c_2 \\ c_3 \end{array}\right)\,,$   and$\displaystyle \quad \vec{x}=\left(\begin{array}{c} 3 \\ -18 \\ -3 \end{array}\right). $

a)
Determine $ \sigma\in\{-1,1\}$ and $ c_2,c_3\in\mathbb{R}$ such that each pair of $ \vec{a}$ , $ \vec{b}$ , $ \vec{c}$ are orthogonal and the corresponding coordinate system obeys the right-hand-rule.
b)
Using the scalar product compute the coefficients $ \alpha$ , $ \beta$ , $ \gamma$ of the linear combination $ \vec{x}=\alpha\vec{a}+\beta\vec{b}+\gamma\vec{c}$ .


Answer:

a)
$ \vec{c}=\left(\rule{0pt}{7ex}\right.$
$ \left.\rule{0pt}{7ex}\right)$
b)
$ \alpha={}$ ,    $ \beta={}$ ,    $ \gamma={}$


  

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  automatically generated: 8/11/2017