Cheatsheet: Stepping up to 3D

What changes from 2D (almost nothing)

2D	3D
2 basis vectors (i-hat, j-hat)	3 basis vectors (i-hat, j-hat, k-hat)
2x2 matrix, columns are 2-vectors	3x3 matrix, columns are 3-vectors
M·v = x·col1 + y·col2	M·v = x·col1 + y·col2 + z·col3
Unit square becomes a parallelogram	Unit cube becomes a parallelepiped

Same definition of linear (origin fixed, grid lines straight/parallel/even), same basis-vector trick, same composition rule. Just one more dimension.

The standard basis

i-hat = [1, 0, 0]   j-hat = [0, 1, 0]   k-hat = [0, 0, 1]

Every 3D vector: [3, 4, 5] = 3·i-hat + 4·j-hat + 5·k-hat.

A 3x3 matrix

[ a  b  c ]   col 1 [a,d,g] = L(i-hat)
[ d  e  f ]   col 2 [b,e,h] = L(j-hat)
[ g  h  i ]   col 3 [c,f,i] = L(k-hat)

Worked examples (applied to [3, 4, 5])

Transformation	Matrix	Result
Rotate 90 deg about z	[[0,-1,0],[1,0,0],[0,0,1]]	[-4, 3, 5] (x,y rotate; z fixed)
Rotate 90 deg about y	[[0,0,1],[0,1,0],[-1,0,0]]	[5, 4, -3] (x,z rotate; y fixed)
Scale (2, 3, 0.5)	[[2,0,0],[0,3,0],[0,0,0.5]]	[6, 12, 2.5] (each axis by its own factor)

In 3D, “rotation” requires naming the axis: rotation about z, y, x are three different transformations.

Sketching the unit cube

1. Read the three columns: where i-hat, j-hat, k-hat land.
2. Plot them as three arrows from the origin.
3. Those are the three edges of the transformed box (parallelepiped).

Columns nearly coplanar => the box squashes nearly flat => the transformation is collapsing space toward a plane or line.

Pitfalls to dodge

• “Rotation” without an axis. In 3D you must name the axis you turn around.
• Expecting new rules. None appear in 3D; it is 2D with one more basis vector.
• Reading rows. Meaning is in the columns: three destination vectors.
• Expecting the cube to stay a cube. Only rigid rotation keeps it a cube; stretch, shear, and flatten reshape it.

Words to use precisely

• k-hat: the third standard basis vector, [0, 0, 1], along z.
• 3x3 matrix: three 3D columns, one each for i-hat, j-hat, k-hat.
• Parallelepiped: the slanted box the unit cube becomes; its edges are the three columns.

The one-line version

3D adds a dimension, not a difficulty: one more basis vector, one more column, one more term, and every 2D rule runs unchanged.