3.13 The scalar product questions for IB Maths AA

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

Find the coordinates of $\mathrm{D}$.

Show that $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a rhombus.

The point $\mathrm{M}$ is the midpoint of $[\mathrm{A}\mathrm{C}]$. Find the coordinates of $\mathrm{M}$ and show that $\mathrm{M}$ is also the midpoint of $[\mathrm{B}\mathrm{D}]$.

Find the value of $\cos \mathrm{A}\widehat{\mathrm{B}}\mathrm{C}$.

Show that the diagonals of $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ are perpendicular.

Hence find the exact area of $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$.

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

Find $\mathrm{A}\widehat{\mathrm{B}}\mathrm{C}$, giving your answer in degrees.

Find the area of triangle ABC.

Using the result $\mathbf{a}\cdot \mathbf{b}=\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, show that $\mathrm{\mid }\mathbf{a}\cdot \mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }$, and state the condition on $\mathbf{a}$ and $\mathbf{b}$ for which equality holds.

Let $\mathbf{a}=\left(\begin{array}{c}2\\ 3\\ 6\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, where $x^2+y^2+z^2=1$.

Show that $2x+3y+6z\le 7$.

Find the values of $x$, $y$ and $z$ for which $2x+3y+6z=7$.

The proof in part (a) assumes that the angle $\theta$ exists, which itself relies on ${\displaystyle \frac{\mathbf{a}\cdot \mathbf{b}}{\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }}}$ lying between $-1$ and $1$. In the rest of the question, you will prove the inequality without using angles.

Let $f(t)=\mathrm{\mid }\mathbf{a}+t\mathbf{b}{\mathrm{\mid }}^2$, where $t\in \mathbb{R}$.

Show that $f(t)=\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2{t}^2+2(\mathbf{a}\cdot \mathbf{b})t+\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2$.

Explain why $f(t)\ge 0$ for all $t\in \mathbb{R}$, and hence show that $(\mathbf{a}\cdot \mathbf{b}{)}^2\le \mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$.

Show that if $(\mathbf{a}\cdot \mathbf{b}{)}^2=\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$, then $\mathbf{a}$ and $\mathbf{b}$ are parallel.

Use the result of part (d) to prove that $\mathrm{\mid }\mathbf{a}+\mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }+\mathrm{\mid }\mathbf{b}\mathrm{\mid }$.

The real numbers $x$, $y$ and $z$ satisfy $x+2y-2z=12$.

By choosing suitable vectors $\mathbf{a}$ and $\mathbf{b}$, find the minimum value of $x^2+y^2+z^2$ and the values of $x$, $y$ and $z$ for which it occurs.

Show that $(p+q+1{)}^2\le 3(p^2+q^2+1)$ for all $p,q\in \mathbb{R}$.

Find $\overrightarrow{\mathrm{O}\mathrm{C}}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Find $\overrightarrow{\mathrm{O}\mathrm{D}}$ in terms of $\mathbf{a}$.

Find $\overrightarrow{\mathrm{A}\mathrm{D}}$ in terms of $\mathbf{a}$.

The point $\mathrm{M}$ is the midpoint of $[\mathrm{C}\mathrm{D}]$. Show that $\overrightarrow{\mathrm{A}\mathrm{M}}={\displaystyle \frac{1}{2}}\mathbf{b}-2\mathbf{a}$.

The line $(\mathrm{A}\mathrm{M})$ meets the line $(\mathrm{B}\mathrm{E})$ at the point $\mathrm{X}$. Find $\overrightarrow{\mathrm{O}\mathrm{X}}$ in terms of $\mathbf{b}$.

Write down the ratio $\mathrm{A}\mathrm{X}:\mathrm{X}\mathrm{M}$.

It is given that $\mathrm{\mid }\mathbf{a}\mathrm{\mid }=4$. Find the exact length of $[\mathrm{A}\mathrm{M}]$.

Find the value of $p$.

Find the size of each of these equal angles.

In the first case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}8\\ 0\end{array}\right)+t\left(\begin{array}{c}0\\ 6\end{array}\right)$, and the patrol boat travels at $10{\text{ km h}}^{-1}$.

Write down the speed of the ship.

Show that the patrol boat meets the ship when $t=1$.

Find the velocity vector of the patrol boat and the bearing on which it travels.

In general, let the ship's position vector be $\mathbf{r}=\mathbf{a}+t\mathbf{u}$, where $\mathrm{\mid }\mathbf{a}\mathrm{\mid }=d>0$ and $\mathrm{\mid }\mathbf{u}\mathrm{\mid }=w>0$. Let $\alpha$ be the angle between $\mathbf{a}$ and $\mathbf{u}$, where $0\le \alpha \le \pi$.

Show that the patrol boat can meet the ship at time $t>0$ if and only if

Suppose that $v>w$. Show that the patrol boat can always intercept the ship, and that there is exactly one possible time of interception.

Suppose now that $v<w$. Show that interception is only possible if $\cos \alpha <0$.

Show that interception is also only possible if $v\ge w\sin \alpha$.

Consider the case $v=w$. Find an expression for the time of interception in terms of $d$, $w$ and $\alpha$, and state the condition on $\alpha$ for interception to be possible.

In the second case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}20\\ 15\end{array}\right)+t\left(\begin{array}{c}-10\\ 0\end{array}\right)$.

Find the minimum speed at which the patrol boat can intercept the ship.

The patrol boat travels at $8{\text{ km h}}^{-1}$. Find the earliest time at which it can intercept the ship.

The patrol boat travels at the minimum speed found in part (f)(i).

Find the time and the point of interception, and show that the velocity of the patrol boat is perpendicular to the initial position vector of the ship.