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

Compute the endpoint after taking 3 steps of $\mathbf{v_1}=(1,-2)$ and 2 steps of $\mathbf{v_2}=(6,-5)$ starting from the origin.

If you take 2 forward steps of $\mathbf{v_1}=(1,-2)$ and then 3 backward steps of $\mathbf{v_2}=(6,-5)$, what point do you reach?

If you first take one step of $\mathbf{v_2}=(6,-5)$, then backtrack along $\mathbf{v_1}=(1,-2)$ $k$ times to reach the point $(2,3)$, find $k$.

Solve for integers $a,b$ such that $a\mathbf{v_1}+b\mathbf{v_2}=(2,3)$ where $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$.

Determine all integer solutions $(a,b)$ such that $a\mathbf{v_1}+b\mathbf{v_2}=(6,-5)$ with $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$.

Can the point $(1,1)$ be reached by integer combinations of $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$? Justify your answer.

List all points that can be reached in exactly 3 forward moves (only positive multiples) using $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$ from the origin.

What is the minimum number of moves (allowing forward or backward steps) needed to reach $(2,3)$ from the origin? Provide the sequence of moves.

Is the point $(3,0)$ reachable using forward or backward moves of $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$? If yes, find one integer combination; if not, explain why.

Find a sequence of moves using exactly 4 forward steps of $\mathbf{v_1}$ or $\mathbf{v_2}$ to reach $(2,3)$, or explain why it is impossible.

Determine whether it is possible to reach $(2,3)$ using exactly 5 moves (forward or backward) along $\mathbf{v_1}$ or $\mathbf{v_2}$. If yes, give the counts of each move.

Explain why $\mathbf{v_1}=(1,-2)$ and $\mathbf{v_2}=(6,-5)$ form a basis for $\mathbb{R}^2$ by computing the determinant of the matrix with columns $\mathbf{v_1},\mathbf{v_2}$.