Equation for Line through Two Points in Complex Plane/Examples/2+i, 3-2i/Standard Form
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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:
- $3 x + y = 7$
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: $70 \ \text {(a)}$