Area of Triangle in Determinant Form/Examples/Vertices at (-4-i), (1+2i), (4-3i)
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Example of Area of Triangle in Determinant Form
Let $T$ be a triangle embedded in the complex plane with vertices at $\paren {-4 - i}, \paren {1 + 2 i}, \paren {4 - 3 i}$.
The area of $T$ is given by:
- $\map \Area T = 17$
Proof
From Area of Triangle in Determinant Form:
$\map \Area T = \dfrac 1 2 \size {\paren {\begin{vmatrix} -4 & -1 & 1 \\ 1 & 2 & 1 \\ 4 & -3 & 1 \\ \end{vmatrix} } }$
\(\ds \map \Area T\) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {\begin{vmatrix}
-4 & -1 & 1 \\ 1 & 2 & 1 \\ 4 & -3 & 1 \\ \end{vmatrix} } }\) |
||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {\paren {-4} \times 2 - \paren {-1} \times 1} - \paren {\paren {-4} \times \paren {-3} - 4 \times \paren {-1} } + \paren {1 \times \paren {-3} - 2 \times 4} }\) | Definition of Determinant | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {-8 - \paren {-1} } - \paren {12 - \paren {-4} } + \paren {-3 - 8} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {-7} - \paren {16} + \paren {-11} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {-34}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 17\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $114$