3.12 Vectors questions for IB Maths AA

Find $\overrightarrow{\mathrm{A}\mathrm{B}}$.

Find $\mathrm{A}\mathrm{B}$.

Write down a unit vector in the direction of $\overrightarrow{\mathrm{A}\mathrm{B}}$.

The point $\mathrm{P}$ lies on the line $(\mathrm{A}\mathrm{B})$ such that $\mathrm{A}\mathrm{P}=14$. Find the two possible sets of coordinates of $\mathrm{P}$.

Find the possible values of $p$.

For the positive value of $p$, find the vector $\mathbf{c}$ such that $\mathrm{\mid }\mathbf{c}\mathrm{\mid }=15$ and $\mathbf{c}$ is in the same direction as $\mathbf{a}$.

Consider a triangle ($n=3$). Show that ${\mathbf{p}}_1={\mathbf{m}}_1-{\mathbf{m}}_2+{\mathbf{m}}_3$.

The midpoints of the sides of a triangle are ${\mathrm{M}}_1(2,1,3)$, ${\mathrm{M}}_2(4,-1,5)$ and ${\mathrm{M}}_3(1,0,2)$. Find the coordinates of ${\mathrm{P}}_1$, ${\mathrm{P}}_2$ and ${\mathrm{P}}_3$.

For any triangle, show that $\overrightarrow{{\mathrm{P}}_1{\mathrm{P}}_2}=2\overrightarrow{{\mathrm{M}}_3{\mathrm{M}}_2}$.

State what this result tells you about the side $[{\mathrm{P}}_1{\mathrm{P}}_2]$ and the line segment $[{\mathrm{M}}_3{\mathrm{M}}_2]$.

Now consider a quadrilateral ($n=4$). Show that ${\mathbf{m}}_1-{\mathbf{m}}_2+{\mathbf{m}}_3-{\mathbf{m}}_4=\mathbf{0}$.

Hence show that ${\mathrm{M}}_1{\mathrm{M}}_2{\mathrm{M}}_3{\mathrm{M}}_4$ is a parallelogram.

The points ${\mathrm{M}}_1(1,2)$, ${\mathrm{M}}_2(4,3)$, ${\mathrm{M}}_3(5,1)$ and ${\mathrm{M}}_4(2,0)$ lie in a plane, with coordinates given in two dimensions.

Verify that these points satisfy the result in part (e).

These points are the midpoints of the sides of a quadrilateral ${\mathrm{P}}_1{\mathrm{P}}_2{\mathrm{P}}_3{\mathrm{P}}_4$, where ${\mathrm{P}}_1$ is the origin. Find the coordinates of ${\mathrm{P}}_2$, ${\mathrm{P}}_3$ and ${\mathrm{P}}_4$.

Explain why the vertices of a quadrilateral are not uniquely determined by the midpoints of its sides.

Now consider a pentagon ($n=5$). Find an expression for ${\mathbf{p}}_1$ in terms of ${\mathbf{m}}_1,{\mathbf{m}}_2,{\mathbf{m}}_3,{\mathbf{m}}_4$ and ${\mathbf{m}}_5$, justifying your answer.

Suggest, with a reason, the values of $n$ for which a polygon with $n$ sides can be reconstructed uniquely from the midpoints of its sides.

The parallelogram is $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$, where $\mathrm{D}$ is diagonally opposite $\mathrm{B}$. Find the coordinates of $\mathrm{D}$.

The fourth vertex of the parallelogram could instead be diagonally opposite $\mathrm{A}$ or diagonally opposite $\mathrm{C}$. Find the coordinates of the fourth vertex in each of these two cases.

Find $\overrightarrow{\mathrm{T}\mathrm{G}}$.

Find the length of the zip wire.

A marker $\mathrm{M}$ is fixed to the wire so that $\mathrm{T}\mathrm{M}:\mathrm{M}\mathrm{G}=3:5$. Find the coordinates of $\mathrm{M}$.

A second marker $\mathrm{N}$ is fixed to the wire $10$ metres from $\mathrm{T}$. Find the coordinates of $\mathrm{N}$.

Consider the case $p={\displaystyle \frac{1}{2}}$ and $q={\displaystyle \frac{1}{3}}$. Find $\overrightarrow{\mathrm{A}\mathrm{Q}}$ and $\overrightarrow{\mathrm{B}\mathrm{P}}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

By writing $\overrightarrow{\mathrm{O}\mathrm{X}}$ in two different ways, find $\overrightarrow{\mathrm{O}\mathrm{X}}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

The line $(\mathrm{O}\mathrm{X})$ meets $[\mathrm{A}\mathrm{B}]$ at the point $\mathrm{R}$. For this case, find $\overrightarrow{\mathrm{O}\mathrm{R}}$ and hence find the ratio $\mathrm{A}\mathrm{R}:\mathrm{R}\mathrm{B}$.

For this case, show that ${\displaystyle \frac{\mathrm{O}\mathrm{P}}{\mathrm{P}\mathrm{A}}}\times {\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}\times {\displaystyle \frac{\mathrm{B}\mathrm{Q}}{\mathrm{Q}\mathrm{O}}}=1$.

Now consider the general case. Show that

Hence show that ${\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}={\displaystyle \frac{q(1-p)}{p(1-q)}}$.

Hence prove that ${\displaystyle \frac{\mathrm{O}\mathrm{P}}{\mathrm{P}\mathrm{A}}}\times {\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}\times {\displaystyle \frac{\mathrm{B}\mathrm{Q}}{\mathrm{Q}\mathrm{O}}}=1$ for all $p$ and $q$.

By considering the case $p=q={\displaystyle \frac{1}{2}}$, deduce that the three medians of a triangle meet at a single point, and find the position vector of this point.

Show that $\mathrm{R}$ is the midpoint of $[\mathrm{A}\mathrm{B}]$ if and only if $(\mathrm{P}\mathrm{Q})$ is parallel to $(\mathrm{A}\mathrm{B})$.