Multiply matching components, then add: Two pipelines leave a pumping station, one heading east-up, one north-down. To compare their directions you don't need a picture — you need one number that captures how much they 'agree'. That number is the scalar (dot) product.
For v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃), multiply the matching components and add:
v·w = v₁w₁ + v₂w₂ + v₃w₃ — the answer is a single number (a scalar), never a vector.
IB-style question — dot product in 3D
A drone's velocity is v = (3, −1, 2) m/s and the wind is w = (4, 5, −1) m/s.
Find v·w.
Step by step
- First write the general rule (multiply matching components, then add).
- Substitute the components.
- Work out each product, then add.
Final answer
v·w = 5 (a scalar). The positive sign tells us the drone and wind broadly point the same way.
It is just a number: The dot product always gives a plain number, with no direction or units of a vector. A positive result means the vectors point broadly the same way; a negative result means broadly opposite; zero means perpendicular (the key fact in §2).
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The dot product also equals |v||w|cosθ: The same number has a second meaning that brings in the angle θ between the two vectors:
v·w = |v| |w| cos θ.
Rearrange to get the angle directly. This is exactly how the IB asks you to find the angle a rope makes with a platform, or the angle at a corner of a triangle ABC.
IB-style question — angle between two vectors
Two support cables from the top of a mast have direction vectors a = (2, 1, 2) and b = (1, −2, 2).
Find the angle between the cables, to the nearest degree.
Step by step
- Start from the angle formula.
- Top: the dot product.
- Bottom: each magnitude (square the components, add, square-root).
- Put it together and take the inverse cosine on the GDC.
Final answer
θ ≈ 64°. The cables open out at about 64° from each other.
Watch the sign of cos θ: If v·w is negative, cos θ is negative, so θ is obtuse (between 90° and 180°). Trust your GDC — cos⁻¹ of a negative number returns the obtuse angle automatically. Don't flip the sign to force an acute answer.