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

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

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 $\mathbf{b}$.

In parallelogram $ABCD$, $\vec{AB}=\langle1,2\rangle$ and $\vec{AD}=\langle2,-1\rangle$. Find $\vec{BD}$ in component form.

Given $\mathbf{p}=(3,2)$ and $\mathbf{q}=(1,-1)$, express the vector from $(0,0)$ to $(8,7)$ as an integer 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 $\mathbf{w}$.

Given moves $\mathbf{u}=(1,2)$ and $\mathbf{v}=(3,5)$ from the origin, show how to reach point $(14,23)$ by an integer combination of $\mathbf{u}$ and $\mathbf{v}$.

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 $\mathbf{a}=(4,1)$ and $\mathbf{b}=(1,3)$. Find integers $x,y$ such that $x\mathbf{a}+y\mathbf{b}=\langle13,6\rangle\,.$

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 $\vec{AC}$.

Point $A$ is at the origin. Using only moves $\mathbf{u}=(2,1)$ and $\mathbf{v}=(-1,2)$, determine whether you can reach $B=(3,5)$. If so, find the combination; if not, explain why.

Given vectors $\mathbf{p}=(1,-2)$ and $\mathbf{q}=(6,-5)$, find integers $x,y$ such that $x\mathbf{p}+y\mathbf{q}=\langle2,3\rangle\,.$