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A parallelogram has Jordan content $∣ \det [v w] ∣$ , equal to base times height when $v \neq 0$

Statement

A parallelogram has Jordan content $∣ \det [v w] ∣$ , equal to base times height when $v \neq 0$ .

More precisely, $P (p; v, w)$ (Parallelograms and triangles in $R^{2}$ ) is Jordan measurable and

$cont (P (p; v, w)) = ∣ \det [v w] ∣,$

including the singular case. If $v \neq 0$ , this value is $∥ v ∥_{2} d (w, R v)$ in the convention of Base and perpendicular height for a chosen side of a plane figure.

Facts & Assumptions

Given: A base point $p \in R^{2}$ and spanning vectors $v, w \in R^{2}$ .

[L1]

For column vectors $v_{1}, \dots, v_{n}$ , the closed parallelepiped is Jordan measurable with content equal to the absolute determinant; this includes the singular case, when the content is zero (The Jordan content of the parallelepiped spanned by the columns of a square real matrix is the absolute value of its determinant).

[L2]

For $v \neq 0$ , $∥ v ∥_{2} d (w, R v) = ∣ \det [v w] ∣$ ( $∥ v ∥_{2} d (w, R v) = ∣ \det [v w] ∣$ for $v \neq 0$ in $R^{2}$ ).

[L3]

Translation preserves Jordan measurability and content (Jordan inner content, outer content, measurability, and content are translation invariant).

Proof

technique · direct

1.1

Specialize [L1] to $n = 2$ and use [L3] to translate the origin-based parallelepiped by $p$ ; it gives Jordan measurability and $cont (P (p; v, w)) = ∣ \det [v w] ∣$ , including dependent or zero spanning vectors.

L1L3

2.1

When $v \neq 0$ , substitute [L2] into the determinant formula of step 1.1 to obtain content equal to base length times perpendicular height.

step 1.1L2∎

Depends on

Used by

The content of a parallelogram from a two-by-two matrix Example

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

W. F. Trench, Introduction to Real Analysis, Theorem 7.3.7 (standard reference, not scraped)
M. E. Taylor, Introduction to Analysis in Several Variables, Proposition 3.1.10 (standard reference, not scraped)

A parallelogram has Jordan content ∣det⁡[v w]∣∣det[v w]∣, equal to base times height when v≠0v=0