Skalarprodukt
\(\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}\cdot\begin{pmatrix}b_1\\b_2\\b_2\end{pmatrix} = a_1b_1 + a_2b_2 + a_3b_3\)
Beispiel.
\(\begin{aligned}&\begin{pmatrix}2\\-3\\-5\end{pmatrix}\cdot\begin{pmatrix}4\\-6\\7\end{pmatrix}\\ =\,& 2\cdot 4 + (-3)\cdot (-6) + (-5)\cdot 7\\ =\,& 8 + 18 - 35 = -9\end{aligned}\)
Vektorprodukt
\(\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}\times\begin{pmatrix}b_1\\b_2\\b_2\end{pmatrix} = \begin{pmatrix}a_2b_3-b_2a_3\\a_3b_1-b_3a_1\\a_1b_2-b_1a_2\end{pmatrix}\)
Beispiel.
\(\begin{aligned}&\begin{pmatrix}2\\-3\\-5\end{pmatrix}\times\begin{pmatrix}4\\-6\\7\end{pmatrix}\\ =&\begin{pmatrix}-3\cdot7-(-6)\cdot(-5)\\-5\cdot4-7\cdot2\\2\cdot(-6)-4\cdot(-3)\end{pmatrix}\\=&\begin{pmatrix}-51\\-34\\0\end{pmatrix}\end{aligned}\)
Winkel \(\alpha\) zwischen den Vektoren \(\vec{a}\) und \(\vec{b}\)
\(\begin{aligned}\alpha = \cos^{-1}\frac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right|\left|\vec{b}\right|}\end{aligned}\)
Beispiel.
Winkel zwischen den Vektoren
\(\begin{pmatrix}2\\-3\\-5\end{pmatrix}\) und \(\begin{pmatrix}4\\-6\\7\end{pmatrix}\)
ist
\(\begin{aligned} & \cos^{-1}\frac{\begin{pmatrix}2\\ -3\\ -5 \end{pmatrix}\cdot\begin{pmatrix}4\\ -6\\ 7 \end{pmatrix}}{\left|\begin{pmatrix}2\\ -3\\ -5 \end{pmatrix}\right|\cdot\left|\begin{pmatrix}4\\ -6\\ 7 \end{pmatrix}\right|}\\ =\, & \cos^{-1}\frac{-9}{\sqrt{2^{2}+\left(-3\right)^{2}+\left(-5\right)^{2}}\cdot\sqrt{4^{2}+\left(-6\right)^{2}+7^{2}}}\\ =\, & \cos^{-1}\frac{-9}{\sqrt{38}\cdot\sqrt{101}}\\ \approx\, & 98,35{^\circ} \end{aligned}\)
