3D vectors — dot product and cross product (IB Math AI HL)

Key ideas

• a·b = a₁b₁ + a₂b₂ + a₃b₃ gives a number. If a·b = 0 (and neither vector is zero), the vectors are perpendicular.
• A negative scalar product means the angle between the vectors is obtuse.
• a × b gives a vector perpendicular to both a and b. Check your answer: its scalar product with each of a and b should be 0.
• |a × b| = |a||b| sin θ is the area of the parallelogram with sides a and b; the triangle with those two sides has half that area.
• The component of a force F in the direction of a unit vector u is F·u, which is how HL AI questions resolve a force along a slope or a cable.
• The order matters for the cross product: b × a = −(a × b). It does not matter for the dot product.

Worked example

Let u = (2, −1, 3) and v = (1, 4, −2). Find the angle between u and v, the vector u × v, and the area of the triangle with sides u and v.

1. u·v = 2 × 1 + (−1) × 4 + 3 × (−2) = 2 − 4 − 6 = −8.
2. |u| = √(4 + 1 + 9) = √14 and |v| = √(1 + 16 + 4) = √21.
3. cos θ = −8 ÷ √(14 × 21) = −8 ÷ √294 ≈ −0.4666, so θ ≈ 117.8° (obtuse, as the negative dot product says).
4. u × v = ((−1)(−2) − 3 × 4, 3 × 1 − 2 × (−2), 2 × 4 − (−1) × 1) = (−10, 7, 9).
5. Check: (−10, 7, 9)·u = −20 − 7 + 27 = 0 and (−10, 7, 9)·v = −10 + 28 − 18 = 0, so it is perpendicular to both.
6. Area of triangle = ½|u × v| = ½√(100 + 49 + 81) = ½√230 ≈ 7.58.

Answer: θ ≈ 117.8°; u × v = (−10, 7, 9); area = ½√230 ≈ 7.58 square units.

Exam-style HL AI questions on this topic

• Scalar Products and Voronoi questions — 32 questions with mark schemes, plus free worked examples
• Vectors and Kinematics questions — 44 questions with mark schemes, plus free worked examples
• All HL AI Geometry and Trigonometry questions by topic

Key formulas for 3D vectors

Vector dot product

a⃗·b⃗ = |a⃗||b⃗|cos θ = a₁ b₁ + a₂ b₂ + a₃ b₃

Vector magnitude

|a⃗| = √(a₁² + a₂² + a₃²)

Angle between two vectors

cos θ = (a⃗·b⃗)/(|a⃗||b⃗|)

Vector equation of a line

r⃗ = a⃗ + t d⃗

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

Do it on your GDC

A GDC is allowed on every AI paper. 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
• Will one phone brand take over? Brand loyalty as a Markov chain
• Crossing a river with a current: fastest or straightest?
• How far does a random walker get? Distance after n steps
• All HL AI IA ideas

Related in Geometry and Trigonometry

• Vector equations of lines in 2D and 3D
• Sinusoidal modelling with seasonal data
• Graph theory — adjacency matrices, spanning trees and route problems

Revise 3D vectors for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI 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 S17 · Problem S73 · Problem S16 · Problem S72. All 22 →

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