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Calculus IIIChapter 1: 3-Dimensional SpaceSection 1-1

The 3-D Coordinate System — Question 8

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Question 8

A transformation TT is performed in this order:

  1. reflect across the yzyz-plane;

  2. then reflect the result across the xyxy-plane.

Tasks

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

  2. Compute T(2,−3,4)T(2,-3,4).

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

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

Original worksheet page 1: question and worked solution for 1-1-008
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Question 8 – Solution

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

See the diagram in the original worksheet below.

Coordinate rule Reflection across the yzyz-plane changes xx to −x-x; reflection across the xyxy-plane changes zz to −z-z. Therefore T(x,y,z)=(−x,y,−z),T(2,−3,4)=(−2,−3,−4).\boxed{T(x,y,z)=(-x,y,-z)},\qquad T(2,-3,4)=\boxed{(-2,-3,-4)}.

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

Distance preservation For P=(x1,y1,z1)P=(x_1,y_1,z_1) and Q=(x2,y2,z2)Q=(x_2,y_2,z_2), d(TP,TQ)2=(−x1+x2)2+(y1−y2)2+(−z1+z2)2=d(P,Q)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 TT twice restores every coordinate, as expected for a half-turn.

Original worksheet page 2: question and worked solution for 1-1-008

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