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

Question 1

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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

Skill question

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