Exercises for Question Type 2: Sketching a path between two points using two given vectors on a grid

Question 1

Can the point (1, 1) be reached by integer combinations of v_{1} = (1, - 2) and v_{2} = (6, - 5) ? Justify your answer.

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Question 2

Determine all integer solutions (a, b) such that a v_{1} + b v_{2} = (6 - 5) with v_{1} = (1 - 2) and v_{2} = (6 - 5) .

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Question 3

Solve for integers a, b such that a v_{1} + b v_{2} = (2, 3) where v_{1} = (1, - 2) and v_{2} = (6, - 5) .

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Question 4

List all points that can be reached from the origin in exactly 3 forward moves using the vectors v_{1} = (1, - 2) and v_{2} = (6, - 5) .

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Question 5

Find the coordinates of the endpoint after taking 3 steps of vector v_{1} = (1 - 2) and 2 steps of vector v_{2} = (6 - 5), starting from the origin.

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Question 6

Is the point (3, 0) reachable using forward or backward moves of v_{1} = (1 - 2) and v_{2} = (6 - 5) ? If yes, find one integer combination; if not, explain why.

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Question 7

Explain why v_{1} = (1 - 2) and v_{2} = (6 - 5) form a basis for R^{2} by computing the determinant of the matrix with columns v_{1}, v_{2} .

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Question 8

Let v_{1} = (11) and v_{2} = (62) be two vectors.

Determine whether it is possible to reach the point (2, 3) using exactly 4 forward steps of v_{1} or v_{2} . If a sequence of moves exists, find it; otherwise, explain why it is impossible.

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Question 9

A particle moves in discrete steps from the origin (0, 0) in a two-dimensional plane. Each step is a move of exactly one vector length in either the forward or backward direction along v_{1} = (11) or v_{2} = (67) .

Determine whether it is possible to reach the point (2, 3) using exactly 5 moves. If it is possible, state the number of forward or backward moves along each vector.

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Question 10

Consider two vectors v_{1} = (1 - 2) and v_{2} = (6 - 5) .

What is the minimum number of moves (allowing forward or backward steps of v_{1} and v_{2}) needed to reach the point (2, 3) from the origin? Provide the sequence of moves.

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Question 11

If you take 2 forward steps of v_{1} = (1 - 2) and then 3 backward steps of v_{2} = (6 - 5), what point do you reach?

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Question 12

If you first take one step of v_{2} = (6 - 5), then backtrack along v_{1} = (1 - 2) k times to reach the point (2, 3), find the value of k .

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IB Mathematics Applications & Interpretation Question Type 2: Sketching a Path Between Two Points Using Two Given Vectors on a Grid Exercises