Vector modelling

Objective. Build complete spatial models from geometric constraints.

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2. Worked example
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Learn: the key idea

Vector modelling builds a full spatial description of a real situation, such as moving objects or geometric shapes, by assigning position vectors, writing equations of motion or lines, and combining constraints to answer questions like collision times or closest approach. A ship's position might be modelled as r = (2,5,0) + t(3,−1,0), with t representing time in hours, allowing prediction of position at any time. Each constraint in the problem, such as a fixed starting point or constant velocity, becomes one piece of the vector equation. Always state clearly what each vector and parameter represents physically before solving.

Worked example

A drone's position is modelled by r = (0,0,10) + t(4,3,0), with t in seconds. Find the vector model's predicted position at t = 2.

1 $Substitute t = 2 into the position component: 0 + 2(4) = 8$
2 $Substitute t = 2 into the next component: 0 + 2(3) = 6$
3 $The height component remains constant at 10$
4 $Combine to give position (8,6,10)$

Answer: (8, 6, 10)

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Common mistake. Students often mix up position and velocity vectors, treating a velocity vector as if it were a starting position. Always keep the fixed starting position and the constant direction (velocity) vector as clearly separate quantities in the model.

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