3 · IB Maths
Vectors and the dot product
Drag the two arrowheads, or use the sliders. The sum follows the parallelogram, and the dot product and the angle between the vectors update live.
Predict first
Start from the default vectors a = (3.0, 2.0) and b = (1.0, 3.0). Change b_x from 1.0 to −2.0 so that b = (−2.0, 3.0). What is the angle between a and b?
a · b = 3 × (−2) + 2 × 3 = 0. A zero dot product between non-zero vectors means cos θ = 0, so θ = 90°.
Check it in the model: Move the b_x slider from 1.0 to −2.0 and read the a · b and Angle between readouts.
a b a + b or a − b. Drag either arrowhead.
cos θ = a · b ÷ (|a||b|)
scalar projection = a · b ÷ |b|
Try this
- Set a = (3.0, 2.0) and b = (−2.0, 3.0). Read the dot product and the angle, then explain why they show the vectors are perpendicular.
- Make b point the opposite way to a, for example b = (−1.5, −1.0). What are the dot product and angle, and what does the sign of a · b tell you?
- Tick the projection option. Keep a fixed and swing b round; at which angles is the scalar projection of a onto b positive, zero and negative?
Common error: a · b is a number, not a vector. Its sign tells you whether the angle is acute (positive), a right angle (zero) or obtuse (negative), and the angle itself needs an inverse cosine of a · b ÷ (|a||b|).
More on 3: Transformations of trigonometric graphs · Solving trigonometric equations · Volume and surface area of 3D solids · Ambiguous case of the sine rule · Vector equations of lines in 3D · Compound angle and double angle identities · Sine and cosine rule · The unit circle · Radians, arcs and sectors · Voronoi diagrams