Vectors Portal

date2026-07-15document_iddoc_612c81be0662ef5fcf66d6151dbeba1ddescriptionベクトル系列の学習順序を示し、成分と演算、幾何学的解釈、位置ベクトル、三次元の積を前提関係に従って接続する。prerequisites座標平面の基本 / 数の四則演算type講義content_typelecturestatusactive

mathvectorportallecture

1Overview

A vector is an algebraic object equipped with addition and scalar multiplication. After coordinates are chosen, vectors can be computed as arrays of components; in geometry, they represent displacement, direction, and magnitude. This portal keeps these interpretations distinct and connects them according to their prerequisites.

2Foundational sequence

Study the following lectures in their _order.json order.

Define vector addition and scalar multiplication componentwise.
Introduce the zero vector, additive inverses, and the standard basis.
Distinguish row vectors from column vectors.
Study geometric vectors, magnitude, dot products, and angles.

data/lecture/math/vector/introduction-to-vectors.lecture.n.md data/lecture/math/vector/zero-opposite-vectors-and-standard-basis.lecture.n.md data/lecture/math/vector/row-and-column-vectors.lecture.n.md data/lecture/math/vector/geometric-vectors-and-dot-products.lecture.n.md

3Points and position vectors

A point represents a location, whereas a vector represents displacement between two points; the two are not identical. After learning this distinction, choose an origin to represent points by position vectors and describe division points and centroids by affine combinations.

data/lecture/math/vector/points-versus-vectors.lecture.n.md data/lecture/math/vector/position-vectors-and-geometry-applications.lecture.n.md data/lecture/math/vector/affine-combinations-and-centroids.lecture.n.md

4Extension to three dimensions

The cross product is defined for real three-dimensional vectors and encodes the area of their parallelogram together with a normal direction determined by operand order. The scalar triple product gives oriented volume. Before studying the cross product, complete the dot product, three-dimensional coordinates, and determinants.

data/lecture/math/linear-algebra/determinants.lecture.n.md data/lecture/math/vector/cross-product-basics.lecture.n.md data/lecture/math/vector/scalar-triple-products-and-volume.lecture.n.md

5Connections to other fields

Linear algebra generalizes these ideas to linear combinations, linear independence, bases, and linear maps.
Coordinate geometry extends descriptions by points and equations to descriptions by position vectors and dot products.
Vector calculus introduces vector-valued functions and differentiation. Single-variable and multivariable calculus are prerequisites.

data/lecture/math/linear-algebra/linear-algebra-portal.lecture.n.md data/lecture/math/geometry/coordinate-geometry-portal.lecture.n.md data/lecture/math/vector-calculus/vector-calculus-portal.lecture.n.md