Vectors in space

The vector AB with initial point A(x₁, y₁, z₁) and endpoint B(x₂, y₂, z₂) has coordinates (x₂ − x₁, y₂ − y₁, z₂ − z₁); equal vectors have equal corresponding coordinates. The length of a(a₁, a₂, a₃) is |a| = √(a₁² + a₂² + a₃²). Addition and multiplication by a number work coordinatewise: a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃), λa = (λa₁, λa₂, λa₃); geometrically the triangle, parallelogram and parallelepiped rules give AB + BC = AC, AB + AD = AC, AB + AD + AA₁ = AC₁. Non-zero a and b are collinear if a = λb, i.e. their coordinates are proportional. With the unit vectors i(1, 0, 0), j(0, 1, 0), k(0, 0, 1) we write a = a₁i + a₂j + a₃k. The dot product is a · b = |a||b| cos φ = a₁b₁ + a₂b₂ + a₃b₃ (a number!). Consequences: cos φ = (a · b)/(|a||b|); a ⊥ b ⇔ a · b = 0; a · a = |a|².