Vocabulary: An ordered triple is a set of three numbers in a specific sequence, often used to represent vectors in three-dimensional space.

Example: Vector v = $a₁ + b₁, a₂ + b₂, a₃ + b₃$ where a and b are the components of two vectors being added.

Definition: A position vector is a vector that represents the position of a point in space relative to the origin of the coordinate system.

Highlight: The position vector OA for point A(x₁, x₂, x₃) is represented as OA = (x₁, x₂, x₃).

Vocabulary: The magnitude of a vector is its length, regardless of its direction.

Example: For vector v = (3, 2, 1), its magnitude |v| = √$3² + 2² + 1²$ = √14 ≈ 3.74

Definition: The distance between two points A and B is equal to the magnitude of the vector AB.

Formula: |AB| = √$(b₁ - a₁)² + (b₂ - a₂)² + (b₃ - a₃)²$

Highlight: Vector addition is commutative: a + b = b + a

Example: For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), their sum is $a₁ + b₁, a₂ + b₂, a₃ + b₃$

Definition: Scalar multiplication involves multiplying each component of a vector by a real number.

Example: For scalar r and vector v = (v₁, v₂, v₃), r · v = (r · v₁, r · v₂, r · v₃)

Highlight: When a vector is multiplied by a negative scalar, its direction is reversed.

Example: For vector v and scalar r > 0, r · v results in a vector in the same direction as v but r times longer.

Formula: The midpoint M of line segment AB is given by OM = 1/2 · (OA + OB)

Vocabulary: The midpoint of a line segment is the point that divides the segment into two equal parts.

Definition: The dot product of two vectors is a scalar value that is the sum of the products of corresponding components.

Formula: For vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), u · v = u₁v₁ + u₂v₂ + u₃v₃

Highlight: Two vectors are orthogonal if and only if their dot product is zero.

Example: Vectors (3, 1, 1) and $-2, 1, 2$ are not orthogonal as their dot product is -3 ≠ 0.