Equation for Line through Two Points in Complex Plane/Examples/2+i, 3-2i/Parametric Form 1
Jump to navigation
Jump to search
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 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds z - \paren {2 + i}\) | \(=\) | \(\ds t \paren {1 - 3 i}\) |
$\blacksquare$
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: $70 \ \text {(a)}$