The 3-D Coordinate System — Question 8

Question 8

A transformation $T$ is performed in this order:

1. reflect across the $yz$-plane;

2. then reflect the result across the $xy$-plane.

Tasks

1. Find a coordinate formula for $T(x,y,z)$.

2. Compute $T(2,-3,4)$.

3. Identify the single rigid motion equivalent to this composition.

4. Prove directly from the three-dimensional distance formula that $T$ preserves the distance between every pair of points.

Question 8 – Solution

Strategy Track which coordinate changes sign under each reflection. Then compare squared distances before and after the transformation.

Coordinate rule Reflection across the $yz$-plane changes $x$ to $-x$; reflection across the $xy$-plane changes $z$ to $-z$. Therefore $$\boxed{T(x,y,z)=(-x,y,-z)},\qquad T(2,-3,4)=\boxed{(-2,-3,-4)}.$$

Geometric interpretation The $y$-coordinate is fixed while both perpendicular coordinates reverse. This is a rotation through $180^\circ$ about the $y$-axis.

Distance preservation For $P=(x_1,y_1,z_1)$ and $Q=(x_2,y_2,z_2)$, $$d(TP,TQ)^2=(-x_1+x_2)^2+(y_1-y_2)^2+(-z_1+z_2)^2=d(P,Q)^2.$$ Both distances are nonnegative, so they are equal.

Verification Applying $T$ twice restores every coordinate, as expected for a half-turn.