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

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:

$3 x + y = 7$

Proof

From Equation of Straight Line in Plane through Two Points, 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 equation:

$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Hence:

\(\ds \dfrac {x - 2} {3 - 2}\)

\(=\)

\(\ds \dfrac {y - 1} {-2 - 1}\)

\(\ds \leadsto \ \ \)

\(\ds x - 2\)

\(=\)

\(\ds \dfrac {y - 1} {-3}\)

\(\ds \leadsto \ \ \)

\(\ds -3x + 6\)

\(=\)

\(\ds y - 1\)

\(\ds \leadsto \ \ \)

\(\ds 3 x + y\)

\(=\)

\(\ds 7\)

$\blacksquare$

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: $70 \ \text {(a)}$