A drone flies from A(1, 2) to B(5, 5), but the mission software asks for the return leg too. Which single formula gives both the outward displacement and its exact reverse?
Start with a prediction. State what a displacement vector needs beyond a distance, and how the return leg differs.
Build the model
A vector carries magnitude and direction. Position vectors locate points from one origin; the displacement 位移向量 from A to B is AB = b − a, and BA = −AB reverses it. Add and subtract components; a scalar multiple keeps a vector parallel.
The scalar product 数量积 a·b = a₁b₁ + a₂b₂ = |a||b| cos θ measures the angle: perpendicular vectors have a·b = 0, and a·a = |a|². Magnitude comes from Pythagoras: |(4, 3)| = 5.
vector: A quantity with magnitude and direction; displacement: The vector from one point to another, b − a; scalar product: The sum of component products, equal to |a||b| cos θ.
Choose evidence that can test it
Keep the order of subtraction: BA = −AB, so swapping endpoints reverses every component. A claimed parallel needs a scalar-multiple relation between the vectors; a sketch alone is not a proof.
For collinearity of three points, show two displacements are parallel AND share a common point; parallel vectors at different locations prove nothing about one line.
Work from known quantities
State the known values and their units. Choose the relation because its assumptions fit this case, then rearrange before substitution.
Known: a = (4, 1) and b = (1, 3). The scalar product is a·b = 4 × 1 + 1 × 3 = 7. The magnitudes are |a| = √17 and |b| = √10, so cos θ = 7/(√17 × √10) ≈ 0.537, giving θ ≈ 57.5°. Since a·b ≠ 0, the vectors are not perpendicular.
Example:
With A(1, 2) and B(5, 5), find the displacement AB, its magnitude and the return displacement, using the same sequence. (Answer: AB = (5, 5) − (1, 2) = (4, 3); |AB| = √(4² + 3²) = 5; BA = (−4, −3), the same magnitude in the opposite direction.)
Check the conclusion and its limits
A vector answers "how far AND which way"; a distance is only a scalar. The scalar product sees direction: it is largest when the vectors align, zero when perpendicular, and negative when they oppose.
Return to the original observation. Explain what the result supports, which conditions it assumes, and one way to test a competing explanation.
Warn:
AB and BA have the same components. This claim is false: BA = −AB reverses every component — from B back to A is (−4, −3), not (4, 3). Only the magnitudes agree.
Key:
Vectors: displacement = end − start, reversed by the return leg; components add; scalar multiples stay parallel; the scalar product gives the angle, with perpendicularity exactly at a·b = 0.
