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§ Vectors

Vectors

§ Vectors

Vectors

R2VG33 min read

Advanced 3D vectors extend traditional vector operations into three-dimensional space, incorporating x, y, and z components. The cross product emerges as a uniquely three-dimensional operation, producing a vector perpendicular to two given vectors with magnitude equal to the area of the parallelogram they form. These operations form the mathematical foundation for describing motion, forces, and geometric relationships in 3D space.

§ 01

Why it matters

Advanced 3D vectors power modern technology across multiple industries. Computer graphics engines use cross products to calculate surface normals for realistic lighting effects in video games and films. Aircraft navigation systems rely on vector calculations to determine flight paths through 3D airspace, with GPS coordinates processed as position vectors. Engineers designing bridges analyse force vectors in three dimensions to ensure structural stability under loads of 50 tonnes or more. Robotics programming uses parametric line equations to plot precise movements through manufacturing assembly lines. Medical imaging systems like MRI scanners reconstruct 3D models from 2D slices using vector mathematics, processing datasets containing millions of coordinate points. These applications highlight why Year 13 students encounter 3D vectors before progressing to university-level physics and engineering courses.

§ 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, 0) and b⃗ = (-2, -1, 1), find a⃗ − b⃗.

Answer: a⃗ − b⃗ = (7, 4, -1)

  1. Add/subtract component-wise → (5 − -2, 3 − -1, 0 − 1) — The difference is found by applying the operation to each component.
  2. Compute → (7, 4, -1) — x: 5 − -2 = 7, y: 3 − -1 = 4, z: 0 − 1 = -1.
Easy§ 02

Find 2·v⃗ for v⃗ = (-5, 2, -5).

Answer: 2·v⃗ = (-10, 4, -10)

  1. Multiply each component by the scalar → (2×-5, 2×2, 2×-5) — Scalar multiplication scales each component by the same factor.
  2. Compute → (-10, 4, -10) — 2×-5 = -10, 2×2 = 4, 2×-5 = -10.
Medium§ 03

Find a⃗ × b⃗ for a⃗ = (5, 2, 5) and b⃗ = (6, 5, 0).

Answer: a⃗ × b⃗ = (-25, 30, 13)

  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 = 2×0 − 5×5 = 0 − 25 = -25 — a₂b₃ − a₃b₂ = 2×0 − 5×5 = -25.
  3. Compute y-component → y = 5×6 − 5×0 = 30 − 0 = 30 — a₃b₁ − a₁b₃ = 5×6 − 5×0 = 30.
  4. Compute z-component → z = 5×5 − 2×6 = 25 − 12 = 13 — a₁b₂ − a₂b₁ = 5×5 − 2×6 = 13.
  5. Combine → a⃗ × b⃗ = (-25, 30, 13) — The cross product vector is perpendicular to both a⃗ and b⃗.
§ 04

Common mistakes

  • Computing cross products incorrectly by treating them like dot products, yielding a scalar like 15 instead of the vector (-25, 30, 13).
  • Confusing the order in cross product calculations, producing a⃗ × b⃗ = (25, -30, -13) when the correct answer is (-25, 30, 13).
  • Forgetting that cross products are only defined in 3D, attempting to calculate (2, 3) × (1, 4) rather than extending to 3D first.
  • Mixing up parametric line notation by writing r = a + bt instead of r = a + tb, leading to incorrect intersection calculations.
§ 05

Frequently asked questions

What is the cross product of two 3D vectors?
The cross product a⃗ × b⃗ produces a vector perpendicular to both original vectors. For a⃗ = (a₁, a₂, a₃) and b⃗ = (b₁, b₂, b₃), the result is (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Its magnitude equals the area of the parallelogram formed by the two vectors.
How do you find the angle between two 3D vectors?
Use the dot product formula: cos θ = (a⃗ · b⃗) / (|a⃗||b⃗|). Calculate the dot product by multiplying corresponding components and adding: a₁b₁ + a₂b₂ + a₃b₃. Divide by the product of both magnitudes, then use inverse cosine to find the angle.
What makes 3D vectors different from 2D vectors?
3D vectors include a z-component, making calculations like magnitude use √(x² + y² + z²) instead of √(x² + y²). The cross product only exists in 3D, producing vectors perpendicular to the original pair. Geometric interpretations expand from areas and angles in planes to volumes and orientations in space.
How do parametric line equations work in 3D?
A parametric line in 3D has the form r⃗ = a⃗ + t·d⃗, where a⃗ is a point on the line, d⃗ is the direction vector, and t is a parameter. Each value of t gives a different point on the line. For example, r⃗ = (2, 1, 3) + t(1, -1, 2) describes a line passing through (2, 1, 3).
When are two 3D vectors perpendicular?
Two vectors are perpendicular when their dot product equals zero. For vectors a⃗ = (3, 4, 0) and b⃗ = (4, -3, 5), calculate a⃗ · b⃗ = 3×4 + 4×(-3) + 0×5 = 12 - 12 + 0 = 0. This zero result confirms perpendicularity, regardless of vector magnitudes.
§ 06

See also

AreaMultiple
§ 06

Where to next?

Prerequisites
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