Area of Quadrilateral in Determinant Form/Examples/Vertices at (2, -1), (4, 3), (-1, 2), (-3, -2)
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Example of Area of Quadrilateral in Determinant Form
Let $Q$ be a quadrilateral embedded in the cartesian plane with vertices at $\tuple {2, -1}$, $\tuple {4, 3}$, $\tuple {-1, 2}$ and $\tuple {-3, -2}$.
The area of $Q$ is given by:
- $\map \Area Q = 18$
Proof
From Area of Quadrilateral in Determinant Form:
\(\ds \map \Area Q\) | \(=\) | \(\ds \dfrac 1 2 \paren {\size {\paren {\begin{vmatrix}
2 & -1 & 1 \\ 4 & 3 & 1 \\ -1 & 2 & 1 \\ \end{vmatrix} } } + \size {\paren {\begin{vmatrix} 2 & -1 & 1 \\ -3 & -2 & 1 \\ -1 & 2 & 1 \\ \end{vmatrix} } } }\) |
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\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {2 \times 3 - 4 \times \paren {-1} } - \paren {2 \times 2 - \paren {-1} \times \paren {-1} } + \paren {4 \times 2 - \paren {-1} \times 3} }\) | Definition of Determinant | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac 1 2 \size {\paren {2 \times \paren {-2} - \paren {-3} \times \paren {-1} } - \paren {2 \times 2 - \paren {-1} \times \paren {-1} } + \paren {\paren {-3} \times 2 - \paren {-1} \times \paren {-2} } }\) | Definition of Determinant | ||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \size {\paren {6 - \paren {-4} } - \paren {4 - 1} + \paren {8 - \paren {-3} } }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac 1 2 \size {\paren {\paren {-4} - 3} - \paren {4 - 1} + \paren {\paren {-6} - 2} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\size {10 - 3 + 11} + \size {-7 - 3 + \paren {-8} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {\size {18} + \size {-18} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 2 \paren {36}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 18\) |
$\blacksquare$
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: $115$