Begründen Sie für zweidimensionale Vektoren die Formel

Question

$$ { \left( \overrightarrow { a } \circ \overrightarrow { b }  \right)  }^{ 2 }+{ \left( det\left( \overrightarrow { a } ,\overrightarrow { b }  \right)  \right)  }^{ 2 }={ \left| \overrightarrow { a }  \right|  }^{ 2 }{ \left| \overrightarrow { b }  \right|  }^{ 2 } $$

Answer 1

$$( \begin{pmatrix} x\\y \end{pmatrix}  * \begin{pmatrix} u\\v \end{pmatrix})^2 +det( \begin{pmatrix} x\\y \end{pmatrix}  , \begin{pmatrix} u\\v \end{pmatrix})^2$$$$=(x*u +y*v)^2 + (x*v-u*y)^2$$$$=x^2*u^2 +2xuyv+y^2*v^2+ x^2*v^2 -2xuyv+y^2*u^2$$$$=x^2*u^2 +y^2*v^2+ x^2*v^2 +y^2*u^2$$$$=(x^2 +y^2)* (v^2 +u^2)$$$$|\begin{pmatrix} x\\y \end{pmatrix}|^2 * |\begin{pmatrix} u\\v \end{pmatrix}|^2$$

Answer 2

Setzt man \(\vec{a}=\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}\) und \(b=\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}\) ein, dann bekommt man $$\left(\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}\circ\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}\right)^{2}+\left(\det\begin{pmatrix}a_{1}&b_1\\a_{2}&b_2\end{pmatrix}\right)^{2}=\left|\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}\right|^{2}\cdot\left|\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}\right|^{2}\text{.}$$ Rechne aus.