Image of a Point in a 3D Line Reflection in 3D

Point $P \rightarrow$ Foot of Perpendicular $F \rightarrow$ Image Point $P' = 2F - P$

Object Point $P(x_0, y_0, z_0)$ P(1.0, 6.0, 3.0)

x-pos: 1.0

y-pos: 6.0

z-pos: 3.0

Line $L: \vec{r} = \vec{a} + \lambda\vec{b}$ 3D Vector Form

Point on line $A(x_1, y_1, z_1)$: (0, 1, 2)

Direction vector $\vec{b}\langle a, b, c\rangle$: ⟨1, 2, 3⟩

Cartesian: $\frac{x - 0}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$

Direction Vector $\vec{b}$:

$b_x$1

$b_y$2

$b_z$3

Derivation & Reflection Live Solved

1. Orthogonality: $\vec{PF} \cdot \vec{b} = 0$
Parameter $\lambda = \frac{(\vec{P} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} = $ 1.00

Foot $F$: F(1.0, 3.0, 5.0)

Image $P' = 2F - P$: P'(1.0, 0.0, 7.0)

Distance $|PF| = |FP'|$: 3.61 units

Perpendicular Construction $P-F-P'$ Right Angle ($\perp$) Indicator at $F$ Projection Drop-Lines to Ground Ground Coordinate Grid

X-Axis

Y-Axis

Z-Axis

Line $L$

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