IB Mathematics Applications & Interpretation 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 $\mathbf{u}=\left(\begin{array}{c}2\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}-1\\ 2\end{array}\right)$, 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 $ABCD$, $\overrightarrow{AB}=\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\overrightarrow{AD}=\left(\begin{array}{c}2\\ -1\end{array}\right)$. Find $\overrightarrow{BD}$ in component form.

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

Let $\mathbf{a}=\left(\begin{array}{c}4\\ 1\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}1\\ 3\end{array}\right)$. Find integers $x$ and $y$ such that $$x\mathbf{a}+y\mathbf{b}=\left(\begin{array}{c}13\\ 6\end{array}\right).$$

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

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

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

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

Given $\mathbf{p}=\left(\genfrac{}{}{0ex}{}{3}{2}\right)$ and $\mathbf{q}=\left(\genfrac{}{}{0ex}{}{1}{-1}\right)$, express the vector $\left(\genfrac{}{}{0ex}{}{8}{7}\right)$ as a linear combination of $\mathbf{p}$ and $\mathbf{q}$.

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

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