Vector-to-vector product

As explained in Geometry for Programmers

A vector product is an operation combining vectors, such as a dot product, cross product, or triple product, to produce a scalar or vector.

Vector products combine vectors to support practical geometric work. The dot product produces a scalar, the cross product produces a vector, and the triple product produces a scalar from three vectors. section 9.0 section 9.1

Their value becomes clearer through geometry: vector products encode angles, projections, areas, and volumes. Understanding that geometry helps turn standard operations into custom algorithms rather than merely calling utility functions. section 9.0 section 9.1

The three main products serve different geometric purposes:

• Dot product: combines two vectors into a scalar and supports geometric measurements such as angles and projections. section 9.0 section 9.1
• Cross product: combines two vectors into a vector; its length represents a parallelogram area in the triangle-distance construction. section 9.1 section 10.1.3
• Triple product: combines three vectors into a scalar whose magnitude represents parallelepiped volume and whose sign can encode sidedness. section 9.0 section 10.1.3

Triangle-mesh distance

For a point p and triangle points p1, p2, and p3, the triple product of (p3-p1), (p2-p1), and (p-p1) gives a signed parallelepiped volume. Dividing by the base area—the length of a cross product—gives the point’s signed distance to the triangle’s plane. If the projection falls outside the triangle, cross-product area calculations provide distances to its edges. section 10.1.3

Figure 10.4 The height of the parallelepiped h is the distance from point p to the plane that p₁, p₂, and p₃ make.

Use the parallelepiped-height figure to connect signed triple-product volume with distance from a point to a triangle’s plane. section 10.1.3

How Geometry for Programmers covers it

How deep it goes

Defines three products, explains their geometric roles, and applies them to plane, triangle, and segment distance calculations.

Examples

Uses translated and scaled point-vector models, parallelepiped heights, triangle planes, and point-to-segment distance calculations.

What it assumes

Uses point vectors, vector addition, and scalar multiplication as geometric groundwork for understanding products.

What you can do after

Build geometric algorithms for plane, triangle, and segment distances, including signed distance calculations.

How it connects

Connects dot product, cross product, triple product, vector addition, scalar multiplication, geometric transformations, and signed distance functions.

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Also explained in

• 

  liveBookmanning.com
  Advanced Algorithms and Data StructuresThe cross product is a vector operation in three-dimensional space whose result is perpendicular to its two operand vectors and whose magnitude depends on their sine-separated area.

Compare how 2 books explain vector-to-vector product →

Related concepts

• vectorkind of · A vector is an ordered collection of components representing an element of a vector space.
• triple productused by · The triple product is the scalar obtained by taking the dot product of one vector with the cross product of two other vectors, equal to the signed volume of their parallelepiped.
• determinantuses · A determinant is a scalar computed from a square matrix that characterizes properties such as invertibility and signed volume scaling.
• point-in-triangle predicateused by · A point-in-triangle predicate determines whether a point lies inside, on the boundary of, or outside a specified triangle.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.