Question Type 1: Formulating path between two points using given vectors on an image Exercises

Point $\text{A}$ is located at the origin. Given the moves $\mathbf{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$, determine whether point $\text{B}(3, 5)$ can be reached from $\text{A}$ using only a combination of these moves. Justify your answer.

In parallelogram $ABCD$, $\vec{AB} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\vec{AD} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$. Find $\vec{BD}$ in component form.

In triangle $ABC$, points $M$ and $N$ are midpoints of $AB$ and $AC$ respectively. Express $\vec{MN}$ in terms of $\vec{AB}$ and $\vec{AC}$.

Let $\mathbf{a}=\begin{pmatrix}4\\1\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}1\\3\end{pmatrix}$. Find integers $x$ and $y$ such that $x\mathbf{a}+y\mathbf{b}=\begin{pmatrix}13\\6\end{pmatrix}\,.$

In triangle $ABC$, point $D$ is the midpoint of $AB$ and point $E$ lies on $AC$ such that $\vec{AE} = 2\vec{AC}$. Express $\vec{DE}$ in terms of $\vec{AB}$ and $\vec{AC}$.

Given the vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$, find the integer values of $m$ and $n$ such that $m\mathbf{u} + n\mathbf{v} = \begin{pmatrix} 14 \\ 23 \end{pmatrix}$.

Given vectors $\mathbf{p}=\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and $\mathbf{q}=\begin{pmatrix} 6 \\ -5 \end{pmatrix}$, find integers $x,y$ such that $x\mathbf{p}+y\mathbf{q}=\begin{pmatrix} 2 \\ 3 \end{pmatrix}\,.$

Given points $A,B,C,D$ such that $\vec{AB}=\mathbf{a}$, $\vec{BC}=\mathbf{b}$ and $\vec{CD}=2\mathbf{a}$, express $\vec{AD}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Given $\mathbf{p}=\binom{3}{2}$ and $\mathbf{q}=\binom{1}{-1}$, express the vector $\binom{8}{7}$ as a linear combination of $\mathbf{p}$ and $\mathbf{q}$.

In a cube with vertex $A$ as the origin, the edge vectors are $\vec{AB}=\mathbf{u}$, $\vec{AD}=\mathbf{v}$ and $\vec{AE}=\mathbf{w}$. Let $G$ be the vertex opposite $A$. Express $\vec{BG}$ in terms of $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$.

In parallelogram ABCD, $\vec{AB}=\mathbf{a}$ and $\vec{AD}=\mathbf{b}$. Express $\vec{AC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.