3D Rotations and Reflections

Linear Algebra (Matrices) · Axiom Academy

LESSON Rotating and Reflecting in 3D The same matrices that spin and flip a cube in space — and why the order you apply them changes everything. Spinning a point about the z -axis turns the x and y coordinates exactly like a 2D rotation while leaving z untouched — height in, height out. The animation applies a turn to a wireframe cube: the top and bottom faces stay level , and every vertex sweeps a quarter-circle in its own horizontal plane. z is fixed; x,y rotate in the plane 2. Reflection Across a Coordinate Plane A reflection sends a shape to its mirror image. Reflecting across the xy -plane keeps x and y but negates z — the matrix is the identity with one diagonal sign flipped. The animation drops the cube straight through the xy -plane to the mirror cube below: same footprint, opposite height. , with x,y unchanged. The flipped coordinate is the one perpendicular to the plane. Reflect twice and you are back where you started: M_ xy ^2 = I , so the reflection undoes itself. 3. Composing Rotations — Order Matters Stacking axis rotations builds a general 3D rotation. To rotate first about z , then about y , you multiply : R_y R_z (rightmost acts first). The animation runs the same cube down two paths — R_y R_z on the left, R_z R_y on the right — from the identical start. They land in different orientations , because matrix multiplication does not commute. apply R_z first, then R_y — the order is baked into the product Worked example — the order changes the answer

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