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:

\(\ds x - 2\) \(=\) \(\ds t \paren {3 - 2}\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds 2 + t\)


\(\ds y - 1\) \(=\) \(\ds t \paren {-2 - 1}\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds 1 - 3 t\)

$\blacksquare$


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