Vector Operations

Given two vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃) in ℝ³, their sum is computed component-wise: u + v = (u₁+v₁, u₂+v₂, u₃+v₃). Geometrically, place the tail of v at the tip of u; the resultant vector points from the tail of u to the tip of v. This is called the triangle rule (or parallelogram rule when you draw both vectors from the same point). Subtraction works identically: u − v = (u₁−v₁, u₂−v₂, u₃−v₃), which geometrically represents the vector from the tip of v to the tip of u when both start at the origin. Vector addition is both commutative (u + v = v + u) and associative ((u + v) + w = u + (v + w)).

Multiplying a vector v = (v₁, v₂, v₃) by a real number (scalar) c gives cv = (cv₁, cv₂, cv₃). Geometrically, this stretches or compresses the vector by the factor |c|; if c < 0, the direction reverses. For example, 3v is three times as long and points in the same direction, while −v is the same length but opposite in direction. The magnitude (length) of a vector is ||v|| = √(v₁² + v₂² + v₃²), so ||cv|| = |c|·||v||. A unit vector v̂ = v/||v|| has length 1 and points in the same direction as v.

The dot product of u and v is the scalar u · v = u₁v₁ + u₂v₂ + u₃v₃. It satisfies the fundamental identity u · v = ||u|| ||v|| cos θ, where θ is the angle between the vectors (0 ≤ θ ≤ π). This lets you find angles directly: cos θ = (u · v)/(||u|| ||v||). Critical cases: if u · v = 0, the vectors are orthogonal (perpendicular); if u · v > 0, the angle is acute; if u · v < 0, the angle is obtuse. The dot product also enables scalar projection: the component of u along v is (u · v)/||v||, and the vector projection of u onto v is proj_v(u) = ((u · v)/||v||²) v. These projections are crucial in decomposing forces, computing work done by a force, and finding distances.

The cross product u × v is defined only in ℝ³ and produces a new vector (not a scalar). Using the determinant formula: u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁). Its magnitude satisfies ||u × v|| = ||u|| ||v|| sin θ, which equals the area of the parallelogram spanned by u and v. The direction of u × v is perpendicular to both u and v, following the right-hand rule: curl the fingers of your right hand from u toward v, and your thumb points in the direction of u × v. Unlike the dot product, the cross product is anti-commutative: u × v = −(v × u). If u and v are parallel, u × v = 0. The cross product is essential in physics for computing torque (τ = r × F) and angular momentum.

• \mathbf{u} \cdot \mathbf{v} — Dot product (dimensionless (or product of input units)): A scalar quantity measuring the degree to which two vectors point in the same direction.
• \|\mathbf{u}\|, \|\mathbf{v}\| — Magnitudes (norms) (same as vector components): The lengths of vectors u and v, computed as the square root of the sum of squared components.
• \theta — Angle between vectors (radians or degrees): The angle formed between the two vectors when placed tail-to-tail at a common origin; ranges from 0 to π.
• u_i, v_i — Vector components (varies): The individual coordinate values of each vector along each axis direction.

Problem. Let u = (3, −1, 2) and v = (1, 4, −2). (a) Compute u + v and 2u − v. (b) Find u · v. (c) Find the angle θ between u and v. (d) Find the vector projection of u onto v.

Answer. u + v = (4, 3, 0); 2u − v = (5, −6, 6); u · v = −5; θ ≈ 106.97°; proj_v(u) = (−5/21, −20/21, 10/21)

• Vector Magnitude and Direction — Understanding how to compute the length and direction of a vector is the prerequisite for all vector operations, particularly the dot product formula involving ||u|| and ||v||.
• Matrix Multiplication — Matrix-vector multiplication generalizes scalar multiplication and the dot product; each row of the product is a dot product of a matrix row with the vector.
• Linear Combinations and Span — Vector addition and scalar multiplication are exactly the two operations used to form linear combinations, which define the span and basis of a vector space.
• Law of Cosines — The geometric interpretation of the dot product formula u·v = ||u||||v||cos θ is a direct algebraic restatement of the Law of Cosines applied to the triangle formed by u, v, and u−v.
• Orthogonality and Projections — The dot product directly defines orthogonality (perpendicularity) and is used to compute vector projections, which are foundational for least-squares methods and Gram-Schmidt orthogonalization.