Rechnen mit Vektoren

$\begin{array}[t]{rll} \vec{a}+\vec{b}=&\begin{pmatrix}1\\1\\1\end{pmatrix}+\begin{pmatrix}-3\\4\\2\end{pmatrix} =&\begin{pmatrix}1+(-3)\\1+4\\2+1\end{pmatrix}=\begin{pmatrix}-2\\5\\3\end{pmatrix} \end{array}$

$\begin{array}[t]{rll} 2\vec{b}+\vec{c}=&2\cdot\begin{pmatrix}-3\\4\\2\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix} =&\begin{pmatrix}2\cdot(-3)\\2\cdot4\\2\cdot2\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix} =&\begin{pmatrix}-6\\8\\4\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix} =&\begin{pmatrix}-6+1\\8+0\\4+0\end{pmatrix} =&\begin{pmatrix}-5\\8\\4\end{pmatrix} \end{array}$

$\begin{array}[t]{rll} 3\vec{a}+2\vec{e}=&3\cdot\begin{pmatrix}1\\1\\1\end{pmatrix}+2\cdot\begin{pmatrix}2\\0\\-10\end{pmatrix} =&\begin{pmatrix}3\\3\\3\end{pmatrix}+\begin{pmatrix}4\\0\\-20\end{pmatrix} \\[5pt] =&\begin{pmatrix}3+4\\3+0\\3+(-20)\end{pmatrix} \\[5pt] =&\begin{pmatrix}7\\3\\-17\end{pmatrix} \end{array}$

$\begin{array}[t]{rll} -5\vec{a}+3\vec{d}=&-5\cdot\begin{pmatrix}1\\1\\1\end{pmatrix}+3\cdot\begin{pmatrix}-4\\1\\-2\end{pmatrix} =&\begin{pmatrix}-5\\-5\\-5\end{pmatrix}+\begin{pmatrix}-12\\3\\-6\end{pmatrix}\\[5pt] =&\begin{pmatrix}-5+(-12)\\-5+3\\-5+(-6)\end{pmatrix}\\[5pt] =&\begin{pmatrix}-17\\-2\\-11\end{pmatrix} \end{array}$

$\begin{array}[t]{rll} 2\vec{c}-3\vec{d}=&2\cdot\begin{pmatrix}1\\0\\0\end{pmatrix}-3\cdot\begin{pmatrix}-4\\1\\-2\end{pmatrix} =&\begin{pmatrix}2\\0\\0\end{pmatrix}-\begin{pmatrix}-12\\3\\-6\end{pmatrix}\\[5pt] =&\begin{pmatrix}2-(-12)\\0-3\\0-(-6)\end{pmatrix}\\[5pt] =&\begin{pmatrix}14\\-3\\6\end{pmatrix} \end{array}$

$\begin{array}[t]{rlll} 4\vec{a}+2\vec{d}=&4\cdot\begin{pmatrix}1\\1\\1\end{pmatrix}+2\cdot\begin{pmatrix}-4\\1\\-2\end{pmatrix} =&\begin{pmatrix}4\\4\\4\end{pmatrix}+\begin{pmatrix}-8\\2\\-4\end{pmatrix} =&\begin{pmatrix}4+(-8)\\4+2\\4+(-4)\end{pmatrix}\\[5pt] =&\begin{pmatrix}-4\\6\\0\end{pmatrix} \end{array}$

$\begin{array}[t]{rlll} 2\vec{b}-\vec{d}=&2\cdot\begin{pmatrix}-3\\4\\2\end{pmatrix}-\begin{pmatrix}-4\\1\\-2\end{pmatrix} =&\begin{pmatrix}-6\\8\\4\end{pmatrix}-\begin{pmatrix}-4\\1\\-2\end{pmatrix}=&\begin{pmatrix}-6-(-4)\\8-1\\4-(-2)\end{pmatrix}\\[5pt] =&\begin{pmatrix}-2\\7\\6\end{pmatrix} \end{array}$

$\begin{array}[t]{rl} \vec{a}+2\vec{b}-\vec{d}=&\begin{pmatrix}1\\1\\1\end{pmatrix}+2\cdot\begin{pmatrix}-3\\4\\2\end{pmatrix}-\begin{pmatrix}-4\\1\\-2\end{pmatrix}\\[5pt] =&\begin{pmatrix}1\\1\\1\end{pmatrix}+\begin{pmatrix}-6\\8\\4\end{pmatrix}-\begin{pmatrix}-4\\1\\-2\end{pmatrix}\\[5pt] =&\begin{pmatrix}1+(-6)-(-4)\\1+8-1\\1+4-(-2)\end{pmatrix}\\[5pt] =&\begin{pmatrix}-1\\8\\7\end{pmatrix} \end{array}$

$\begin{array}[t]{rl} 4\vec{b}+3\vec{c}-6\vec{a}=&4\cdot\begin{pmatrix}-3\\4\\2\end{pmatrix}+3\cdot\begin{pmatrix}1\\0\\0\end{pmatrix}-6\cdot\begin{pmatrix}1\\1\\1\end{pmatrix} \\[5pt] =&\begin{pmatrix}-12\\16\\8\end{pmatrix}+\begin{pmatrix}3\\0\\0\end{pmatrix}-\begin{pmatrix}6\\6\\6\end{pmatrix}\\[5pt] =&\begin{pmatrix}-12+3-6\\16+0-6\\8+0-6\end{pmatrix}\\[5pt] =&\begin{pmatrix}-15\\10\\2\end{pmatrix} \end{array}$

$\begin{array}[t]{rl} \vec{a}+\vec{b}+\vec{c}+\vec{d}-5\vec{e}=&\begin{pmatrix}1\\1\\1\end{pmatrix}+\begin{pmatrix}-3\\4\\2\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix}+\begin{pmatrix}-4\\1\\-2\end{pmatrix}-5\cdot\begin{pmatrix}2\\0\\-10\end{pmatrix}\\[5pt] =&\begin{pmatrix}1\\1\\1\end{pmatrix}+\begin{pmatrix}-3\\4\\2\end{pmatrix}+\begin{pmatrix}1\\0\\0\end{pmatrix}+\begin{pmatrix}-4\\1\\-2\end{pmatrix}-\begin{pmatrix}10\\0\\-50\end{pmatrix}\\[5pt] =&\begin{pmatrix}1+(-3)+1+(-4)-10\\1+4+0+1-0\\1+2+0+(-2)-(-50)\end{pmatrix}\\[5pt] =&\begin{pmatrix}-15\\6\\51\end{pmatrix} \end{array}$

$\begin{array}[t]{rll} r\cdot\vec{a}+(r-1)\cdot\vec{b}=&r\cdot\begin{pmatrix}2\\-2\\0\end{pmatrix}+(r-1)\cdot\begin{pmatrix}2\\-4\\7\end{pmatrix}&\scriptsize\mid\; \text{Ausmultiplizieren} \\[5pt] =&r\cdot\begin{pmatrix}2\\-2\\0\end{pmatrix}+r\cdot\begin{pmatrix}2\\-4\\7\end{pmatrix}-1\cdot\begin{pmatrix}2\\-4\\7\end{pmatrix} \\[5pt] =&\begin{pmatrix}2r\\-2r\\0\end{pmatrix}+\begin{pmatrix}2r\\-4r\\7r\end{pmatrix}-\begin{pmatrix}2\\-4\\7\end{pmatrix} \\[5pt] =&\begin{pmatrix}2r+2r\\-2r+(-4r)\\0+7r\end{pmatrix}-\begin{pmatrix}2\\-4\\7\end{pmatrix} \\[5pt] =&\begin{pmatrix}4r\\-6r\\7r\end{pmatrix}-\begin{pmatrix}2\\-4\\7\end{pmatrix} \\[5pt] =&r\cdot\begin{pmatrix}4\\-6\\7\end{pmatrix}-\begin{pmatrix}2\\-4\\7\end{pmatrix} \end{array}$

$\begin{array}[t]{rl} s\cdot(\vec{d}+\vec{c})+2s\cdot\vec{a}=&s\cdot\left(\begin{pmatrix}4\\-5\\3\end{pmatrix}+\begin{pmatrix}1\\1\\3\end{pmatrix}\right)+2s\cdot\begin{pmatrix}2\\-2\\0\end{pmatrix} \\[5pt] =&s\cdot\begin{pmatrix}4+1\\-5+1\\3+3\end{pmatrix}+2s\cdot\begin{pmatrix}2\\-2\\0\end{pmatrix} \\[5pt] =&s\cdot\begin{pmatrix}5\\-4\\6\end{pmatrix}+2s\cdot\begin{pmatrix}2\\-2\\0\end{pmatrix} \\[5pt] =&\begin{pmatrix}5s\\-4s\\6s\end{pmatrix}+\begin{pmatrix}4s\\-4s\\0\end{pmatrix} \\[5pt] =&\begin{pmatrix}5s+4s\\-4s+(-4s)\\6s+0\end{pmatrix} \\[5pt] =&\begin{pmatrix}9s\\-8s\\6s\end{pmatrix} \\[5pt] =&s\cdot\begin{pmatrix}9\\-8\\6\end{pmatrix} \end{array}$

$\begin{array}[t]{rl} r\cdot(\vec{e}-\vec{b})+(r+s)\cdot(\vec{a}-\vec{c})=&r\cdot\left(\begin{pmatrix}-3\\8\\3\end{pmatrix}-\begin{pmatrix}2\\-4\\7\end{pmatrix}\right)+(r+s)\cdot\left(\begin{pmatrix}2\\-2\\0\end{pmatrix}-\begin{pmatrix}1\\1\\3\end{pmatrix}\right) \\ =&r\cdot\begin{pmatrix}-3-2\\8-(-4)\\3-7\end{pmatrix}+(r+s)\cdot\begin{pmatrix}2-1\\-2-1\\0-3\end{pmatrix} \\ =&r\cdot\begin{pmatrix}-5\\12\\-4\end{pmatrix}+(r+s)\cdot\begin{pmatrix}1\\-3\\-3\end{pmatrix}\qquad\scriptsize\mid\;\text{Ausmultiplizieren} \\ =&r\cdot\begin{pmatrix}-5\\12\\-4\end{pmatrix}+r\cdot\begin{pmatrix}1\\-3\\-3\end{pmatrix}+s\cdot\begin{pmatrix}1\\-3\\-3\end{pmatrix} \\ =&\begin{pmatrix}-5r\\12r\\-4r\end{pmatrix}+\begin{pmatrix}1r\\-3r\\-3r\end{pmatrix}+\begin{pmatrix}1s\\-3s\\-3s\end{pmatrix} \\ =&\begin{pmatrix}-5r+1r\\12r-3r\\-4r-3r\end{pmatrix}+\begin{pmatrix}1s\\-3s\\-3s\end{pmatrix} \\ =&\begin{pmatrix}-4r\\9r\\-7r\end{pmatrix}+\begin{pmatrix}1s\\-3s\\-3s\end{pmatrix} \\ =&r\cdot\begin{pmatrix}-4\\9\\-7\end{pmatrix}+s\cdot\begin{pmatrix}1\\-3\\-3\end{pmatrix} \end{array}$