3D vectors — dot product, cross product, and applications (IB Math AA HL)

Key ideas

• Scalar product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. Use it for the angle between two vectors; a·b = 0 means they are perpendicular.
• Vector product: a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), a vector perpendicular to both a and b.
• |a × b| = |a||b| sin θ is the area of the parallelogram on a and b; half of it is the area of the triangle.
• a × b = −(b × a), and a × b = 0 when a and b are parallel.
• The vector product of two direction vectors in a plane gives the plane's normal vector.

Worked example

a = (1, 2, 3) and b = (2, −1, 1). Find a × b and the area of the triangle with sides a and b.

1. a × b = (2 × 1 − 3 × (−1), 3 × 2 − 1 × 1, 1 × (−1) − 2 × 2) = (5, 5, −5).
2. Check: (5, 5, −5)·a = 5 + 10 − 15 = 0 and (5, 5, −5)·b = 10 − 5 − 5 = 0, so it is perpendicular to both.
3. |a × b| = √(25 + 25 + 25) = √75 = 5√3.
4. Area of the triangle = ½ × 5√3 = 5√3/2 ≈ 4.33.

Answer: a × b = (5, 5, −5), and the triangle's area is 5√3/2 ≈ 4.33.

Exam-style HL AA questions on this topic

• Planes questions — 47 questions with mark schemes, plus free worked examples
• Vector Lines questions — 53 questions with mark schemes, plus free worked examples
• All HL AA Geometry and Trigonometry questions by topic

Key formulas for 3D vectors

Vector magnitude

|a⃗| = √(a₁² + a₂² + a₃²)

Vector equation of a line

r⃗ = a⃗ + t d⃗

In the notation of the IB formula booklet. All HL AA formulas · see them used in worked examples

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Vectors on your calculator
• Try it: vector calculator (free, shows the working)

Maths IA ideas that use 3d vectors

• How are letters drawn on a screen? Bézier curves
• Crossing a river with a current: fastest or straightest?
• The hidden angles of a hip roof: 3D trigonometry
• How far does a random walker get? Distance after n steps
• All HL AA IA ideas

Related in Geometry and Trigonometry

• Lines and planes in 3D — intersections and distances
• Vector proofs and geometric arguments
• Compound and double-angle formulas

Read more on the blog

• IB Maths AA HL: Vectors Practice Questions with Full Mark Schemes

Revise 3D vectors for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S21 · Problem S77 · Problem S20 · Problem S76. All 22 →

Formulas in this topic: Magnitude of a vector · Scalar (dot) product. All formulas A–Z