Point of Perpendicular Intersection on Real Line from Points in Complex Plane/Examples/a = 1-3i, b = -3+4i

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Examples of Point of Perpendicular Intersection on Real Line from Points in Complex Plane

Let $a, b \in \C$ be complex numbers represented by the points $A$ and $B$ respectively in the complex plane.

Let $x \in \R$ be a real number represented by the point $X$ on the real axis such that $AXB$ is a right triangle with $X$ as the right angle.

Let $a = 1 - 3 i, b = -3 + 4 i$.

The point $X$ on the positive half of the real axis is at:

$x = 3$


Proof

From Point of Perpendicular Intersection on Real Line from Points in Complex Plane:

$x = \dfrac {a_x + b_x \pm \sqrt {a_x^2 + b_x^2 - 2 a_x b_x - 4 a_y b_y} } 2$

Setting $a = 1 - 3 i, b = -3 + 4 i$:


\(\ds x\) \(=\) \(\ds \dfrac {1 + \paren {-3} \pm \sqrt {1^2 + \paren {-3}^2 - 2 \times 1 \times \paren {-3} - 4 \times \paren {-3} \times 4} } 2\)
\(\ds \) \(=\) \(\ds \dfrac {-2 \pm \sqrt {10 + 6 + 48} } 2\)
\(\ds \) \(=\) \(\ds \dfrac {-2 \pm \sqrt {64} } 2\)
\(\ds \) \(=\) \(\ds -1 \pm 4\)

It is the positive solution that is needed, so:

$x = 3$

$\blacksquare$


Sources