Equation for Line through Two Points in Complex Plane/Examples/2+i, 3-2i/Parametric Form 1

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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 equation:

$z - \paren {2 + i} = t \paren {1 - 3 i}$


Proof

From Equation for Line through Two Points in Complex Plane: Parametric Form $1$, a straight line $L$ passing through $2$ points $z_1$ and $z_2$ has the equation:

$z = z_1 + t \paren {z_2 - z_1}$


Hence:

\(\ds z\) \(=\) \(\ds \paren {2 + i} + t \paren {\paren {3 - 2 i} - \paren {2 + i} }\)
\(\ds \) \(=\) \(\ds \paren {2 + i} + t \paren {1 - 3 i}\) Definition of Complex Subtraction
\(\ds \leadsto \ \ \) \(\ds z - \paren {2 + i}\) \(=\) \(\ds t \paren {1 - 3 i}\)

$\blacksquare$


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