引用:
原帖由 mandy 於 2012-5-5 12:38 PM 發表 
謝謝老師!! 我想起來了, 以前高中時, 的確學過, 是太久沒算了, 謝謝老師 !!
我認為如果是用公式五階的行列式計算四面體的體積, 不見得快, 因為降階從五階降到四階, 再從四階降到三階, 計算就很大,
老師認為呢 ? ...
用#40 彬爸文中的符號
\(\displaystyle BC=a,CA=b,AB=c,DA=\alpha,DB=\beta,DC=\gamma \)
\(\displaystyle 288V^2=\left |
\begin {array} {ccccc}
0 & 1 & 1 & 1 & 1 \\
1 & 0 & c^2 & b^2 & \alpha^2 \\
1 & c^2 & 0 & a^2 & \beta^2 \\
1 & b^2 & a^2 & 0 & \gamma^2 \\
1 & \alpha^2 & \beta^2 & \gamma^2 & 0 \\
\end {array} \right |
\)
\(\displaystyle =\left |
\begin {array} {ccccc}
0 & 1 & 1 & 1 & 1 \\
0 & -\alpha^2 & c^2-\beta^2 & b^2-\gamma^2 & \alpha^2 \\
0 & c^2-\alpha^2 & -\beta^2 & a^2-\gamma^2 & \beta^2 \\
0 & b^2-\alpha^2 & a^2-\beta^2 & -\gamma^2 & \gamma^2 \\
1 & \alpha^2 & \beta^2 & \gamma^2 & 0 \\
\end {array} \right |
\)
\(\displaystyle =\left |
\begin {array} {cccc}
1 & 1 & 1 & 1 \\
-\alpha^2 & c^2-\beta^2 & b^2-\gamma^2 & \alpha^2 \\
c^2-\alpha^2 & -\beta^2 & a^2-\gamma^2 & \beta^2 \\
b^2-\alpha^2 & a^2-\beta^2 & -\gamma^2 & \gamma^2 \\
\end {array} \right |
\)
\(\displaystyle =\left |
\begin {array} {cccc}
0 & 0 & 0 & 1 \\
-2\alpha^2 & c^2-\beta^2-\alpha^2 & b^2-\gamma^2-\alpha^2 & \alpha^2 \\
c^2-\alpha^2-\beta^2 & -2\beta^2 & a^2-\gamma^2-\beta^2 & \beta^2 \\
b^2-\alpha^2-\gamma^2 & a^2-\beta^2-\gamma^2 & -2\gamma^2 & \gamma^2 \\
\end {array} \right |
\)
\(\displaystyle =-\left |
\begin {array} {ccc}
-2\alpha^2 & c^2-\beta^2-\alpha^2 & b^2-\gamma^2-\alpha^2 \\
c^2-\alpha^2-\beta^2 & -2\beta^2 & a^2-\gamma^2-\beta^2 \\
b^2-\alpha^2-\gamma^2 & a^2-\beta^2-\gamma^2 & -2\gamma^2 \\
\end {array} \right |
\)
\(\displaystyle =\left |
\begin {array} {ccc}
2\alpha^2 & \alpha^2+\beta^2-c^2 & \alpha^2+\gamma^2-b^2 \\
\alpha^2+\beta^2-c^2 & 2\beta^2 & \beta^2+\gamma^2-a^2 \\
\alpha^2+\gamma^2-b^2 & \beta^2+\gamma^2-a^2 & 2\gamma^2 \\
\end {array} \right |
\)
這就得到 彬爸 所PO的公式。
至於你說的計算難度問題，的確，五階比三階難算得多，
實際運用時，就憑你的記憶和計算能力吧。(感謝寸絲老師~~)
