3.14 Vector equations of lines questions for IB Maths AA

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

Show that $\mathrm{A}\widehat{\mathrm{B}}\mathrm{C}=90\mathrm{°}$.

ABCD is a rectangle. Find the coordinates of D.

Find the cosine of the acute angle between the diagonals [AC] and [BD].

The point P lies on [AC] such that [BP] is perpendicular to [AC].

Find the coordinates of P.

Find the exact length of [BP].

Find $\overrightarrow{\mathrm{O}\mathrm{P}}$ and $\overrightarrow{\mathrm{P}\mathrm{Q}}$.

Find the total distance flown by the drone from $\mathrm{O}$ to $\mathrm{Q}$.

Find the straight-line distance from $\mathrm{O}$ to $\mathrm{Q}$.

Find a unit vector in the direction of $\overrightarrow{\mathrm{P}\mathrm{Q}}$.

On the second stage of its flight the drone flies from $\mathrm{P}$ towards $\mathrm{Q}$ at a constant speed of $12.5$ m s${}^{-1}$. Find the coordinates of the drone $6$ seconds after it leaves $\mathrm{P}$.

The top of a mast is at the point $\mathrm{T}(200,10,40)$. Find the length of time for which the drone is within $50$ metres of $\mathrm{T}$ during its flight from $\mathrm{P}$ to $\mathrm{Q}$.

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.

Find a vector equation of $L$.

Determine whether the point $\mathrm{C}(7,9,-6)$ lies on $L$.

Consider the line $L_2$.

Write down the coordinates of the point on $L_2$ for which $\lambda =3$.

Verify that this point lies on $S$.

Show that every point of $L_a$ lies on $S$.

Show that $L_1$ and $L_2$ are skew.

Show that, for $a\mathrm{\ne }c$, the lines $L_a$ and $L_c$ are skew.

Show that $L_a$ and $M_b$ intersect, and find the coordinates of their point of intersection in terms of $a$ and $b$.

The point $\mathrm{P}(p,q,pq)$ lies on $S$.

Write down the value of $a$ and the value of $b$ for which both $L_a$ and $M_b$ pass through P.

Let $\theta$ be the acute angle between $L_p$ and $M_q$.

Show that $\cos \theta ={\displaystyle \frac{\mathrm{\mid }pq\mathrm{\mid }}{\sqrt{(1+p^2)(1+q^2)}}}$.

Hence show that $L_p$ and $M_q$ are perpendicular if and only if P lies on the $x$-axis or on the $y$-axis.

The plane $\mathrm{\Pi }$ contains both $L_p$ and $M_q$.

Show that the Cartesian equation of $\mathrm{\Pi }$ is $qx+py-z=pq$.

By substituting $z=xy$ into the equation of $\mathrm{\Pi }$, show that the only points of $S$ that lie in $\mathrm{\Pi }$ are the points of $L_p$ and $M_q$.