3×3 parallelepiped story

In $\mathbb{R}^3$, columns of $A$ span a parallelepiped. Interpret $\mathbf{b}$ inside that solid; ratios of signed volumes pin coordinates . The scalar triple product $\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w})$ equals oriented volume.

Right-hand orientation keeps signs aligned with column order. If columns are nearly dependent, volumes shrink and Cramer ratios become ill-conditioned: tiny denominator errors explode in the division.

Picture $\mathbf{b}$ decomposed inside the parallelepiped: each coordinate measures how much of $\mathbf{b}$ lies along one column direction relative to the full oriented volume .

Reflection inside column data flips sign of determinants because orientation reverses. That sign flip propagates to every Cramer coordinate unless numerator and denominator flip together.

Cramer's rule does not apply directly to nonsquare systems without an invertible square reduction or a different solution concept such as least squares.

In 3×3, each numerator determinant is itself a scalar triple product in disguise, linking this chapter back to cross products and signed volumes from Chapters 10 and 11.

Draw one parallelepiped with labeled columns and shade the face used in a numerator determinant; the picture is the proof template.