Equation for Line through Two Points in Complex Plane/Examples/2+i, 3-2i/Parametric Form 2
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Example of Use of Equation for Line through Two Points in Complex Plane
Let $L$ be a straight line through $2 + i$ and $3 - 2 i$ in the complex plane.
Then $L$ can be expressed by the equations:
- $x = 2 + t, y = 1 - 3 t$
Proof
From Equation for Line through Two Points in Complex Plane: Parametric Form $2$, a straight line $L$ passing through $2$ points $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ has the equations:
\(\ds x - x_1\) | \(=\) | \(\ds t \paren {x_2 - x_1}\) | ||||||||||||
\(\ds y - y_1\) | \(=\) | \(\ds t \paren {y_2 - y_1}\) |
Hence:
\(\ds x - 2\) | \(=\) | \(\ds t \paren {3 - 2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 2 + t\) |
\(\ds y - 1\) | \(=\) | \(\ds t \paren {-2 - 1}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 1 - 3 t\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $70 \ \text {(a)}$