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§ Vectors

Vectors

§ Vectors

Vectors

R2VG33 min read

Advanced 3D vectors extend basic vector operations into three-dimensional space, incorporating x, y, and z components. These vectors support operations like dot products, cross products, and parametric equations that describe lines and planes in 3D coordinate systems. The cross product uniquely produces a vector perpendicular to both input vectors, while the dot product yields a scalar measuring alignment between vectors.

§ 01

Why it matters

Advanced 3D vectors form the mathematical foundation for computer graphics, robotics, and physics simulations. Video game engines use cross products to calculate surface normals for lighting effects, while dot products determine viewing angles and collision detection. In aerospace engineering, vectors describe flight paths and satellite orientations in 3D space. CAD software relies on parametric line equations to model complex 3D objects and calculate intersections between surfaces. Physics uses vector operations to analyze forces in 3D structures, from bridge designs to molecular modeling. The dot product appears in machine learning algorithms for similarity calculations, while cross products help determine torque and angular momentum in mechanical systems operating across multiple dimensions.

§ 02

How to solve vectors

Advanced Vectors

  • Magnitude: |v| = √(x² + y²) for 2D, √(x² + y² + z²) for 3D.
  • Dot product: a·b = a₁b₁ + a₂b₂ + a₃b₃. Equals 0 when the vectors are perpendicular.
  • Unit vector: v / |v|. Has length 1 in the same direction.
  • Angle between vectors: cos θ = (a·b) / (|a||b|).

Example: For a = (3, 4): |a| = √(9 + 16) = 5. Unit vector: (35, 45).

§ 03

Worked examples

Beginner§ 01

Given a⃗ = (-5, -3, -5) and b⃗ = (-1, 0, -4), find a⃗ − b⃗.

Answer: a⃗ − b⃗ = (-4, -3, -1)

  1. Add/subtract component-wise → (-5 − -1, -3 − 0, -5 − -4) — The difference is found by applying the operation to each component.
  2. Compute → (-4, -3, -1) — x: -5 − -1 = -4, y: -3 − 0 = -3, z: -5 − -4 = -1.
Easy§ 02

Find 4·v⃗ for v⃗ = (5, -4, 4).

Answer: 4·v⃗ = (20, -16, 16)

  1. Multiply each component by the scalar → (4×5, 4×-4, 4×4) — Scalar multiplication scales each component by the same factor.
  2. Compute → (20, -16, 16) — 4×5 = 20, 4×-4 = -16, 4×4 = 16.
Medium§ 03

Find a⃗ × b⃗ for a⃗ = (-3, -2, 4) and b⃗ = (0, -4, 2).

Answer: a⃗ × b⃗ = (12, 6, 12)

  1. Use the cross product formula (determinant method) → a⃗ × b⃗ = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) — The cross product is computed using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row.
  2. Compute x-component → x = -2×2 − 4×-4 = -4 − -16 = 12 — a₂b₃ − a₃b₂ = -2×2 − 4×-4 = 12.
  3. Compute y-component → y = 4×0 − -3×2 = 0 − -6 = 6 — a₃b₁ − a₁b₃ = 4×0 − -3×2 = 6.
  4. Compute z-component → z = -3×-4 − -2×0 = 12 − 0 = 12 — a₁b₂ − a₂b₁ = -3×-4 − -2×0 = 12.
  5. Combine → a⃗ × b⃗ = (12, 6, 12) — The cross product vector is perpendicular to both a⃗ and b⃗.
§ 04

Common mistakes

  • Computing cross products incorrectly by mixing up the component formula, such as calculating (-3, -2, 4) × (0, -4, 2) = (-12, 6, 12) instead of (12, 6, 12) due to sign errors in the determinant expansion.
  • Confusing dot product and cross product results, expecting the cross product to produce a scalar like the dot product, when (-3, -2, 4) · (0, -4, 2) = 16 (scalar) but (-3, -2, 4) × (0, -4, 2) = (12, 6, 12) (vector).
  • Incorrectly applying the 2D magnitude formula √(x² + y²) to 3D vectors instead of √(x² + y² + z²), calculating |(-5, -3, -5)| = √(25 + 9) = √34 instead of √(25 + 9 + 25) = √59.
§ 05

Frequently asked questions

What is the difference between dot product and cross product in 3D vectors?
The dot product produces a scalar (single number) that measures how aligned two vectors are, calculated as a₁b₁ + a₂b₂ + a₃b₃. The cross product produces a new vector perpendicular to both input vectors, calculated using the determinant method. For vectors (3, 4, 0) and (1, 2, 5), the dot product equals 11, while the cross product equals (20, -15, 2).
How do you find the angle between two 3D vectors?
Use the formula cos θ = (a·b) / (|a||b|), where a·b is the dot product and |a|, |b| are the magnitudes. For vectors (1, 2, 2) and (2, 1, 0), calculate: dot product = 4, magnitudes are 3 and √5, so cos θ = 4/(3√5) ≈ 0.596, giving θ ≈ 53.4°.
What does it mean when the cross product of two vectors equals zero?
When a × b = (0, 0, 0), the vectors are parallel or antiparallel (pointing in opposite directions). This occurs because the cross product magnitude equals |a||b|sin θ, and sin θ = 0 when θ = 0° or 180°. For example, (2, 4, 6) × (1, 2, 3) = (0, 0, 0) because the second vector is half the first.
How do you check if two 3D vectors are perpendicular?
Two vectors are perpendicular when their dot product equals zero. Calculate a·b = a₁b₁ + a₂b₂ + a₃b₃, and if the result is 0, the vectors form a 90° angle. For instance, vectors (1, 2, -1) and (3, -1, 1) are perpendicular because their dot product is 3 - 2 - 1 = 0.
What is a unit vector and how do you find it?
A unit vector has magnitude 1 and points in the same direction as the original vector. Find it by dividing each component by the vector's magnitude: û = v/|v|. For vector (3, 4, 0) with magnitude 5, the unit vector is (3/5, 4/5, 0) = (0.6, 0.8, 0). Unit vectors are essential for representing pure direction without magnitude.
§ 06

See also

MultipleMean
§ 06

Where to next?

Prerequisites
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