IB Maths AA HL – Topic 3: Geometry and Trigonometry

Let $\mathbf{a} = 2\mathbf{i} – \mathbf{j} + 3\mathbf{k}$ and $\mathbf{b} = \mathbf{i} + 4\mathbf{j} – 2\mathbf{k}$.(a) Find $\mathbf{a}\cdot\mathbf{b}$.(b) Find $\mathbf{a}\times\mathbf{b}$.(c) Hence find the area of the triangle with sides $\mathbf{a}$ and $\mathbf{b}$.

Show that $\tan 75^\circ = 2 + \sqrt{3}$.

Prove the identity $\dfrac{\sec\theta}{\sec\theta – \cos\theta} \equiv \csc^2\theta$.

Solve $\sec^2 x = 3\tan x – 1$ for $0 \leq x < 2\pi$, giving answers in terms of $\pi$ where possible and otherwise in the form $\arctan k$ (plus multiples of $\pi$).

The line $L$ has equation $\mathbf{r} = \begin{pmatrix}1\\2\\-1\end{pmatrix} + t\begin{pmatrix}2\\1\\3\end{pmatrix}$ and the plane $\Pi$ has equation $x + 2y – z = 2$.(a) Find the coordinates of the point where $L$ meets $\Pi$.(b) Find the acute angle between $L$ and $\Pi$.

Points $A(1,0,2)$, $B(3,1,0)$ and $C(0,2,1)$ lie in the plane $\Pi$.(a) Find $\overrightarrow{AB}\times\overrightarrow{AC}$.(b) Hence find a Cartesian equation of $\Pi$.(c) Find the distance from the point $D(4,1,5)$ to $\Pi$.

Express $\sin(x + 30^\circ)$ in terms of $\sin x$ and $\cos x$, and hence solve $\sin(x + 30^\circ) = 2\cos x$ for $0^\circ \leq x < 360^\circ$.

The lines $L_1: \mathbf{r} = \begin{pmatrix}1\\2\\3\end{pmatrix} + s\begin{pmatrix}1\\-1\\2\end{pmatrix}$ and $L_2: \mathbf{r} = \begin{pmatrix}0\\1\\-1\end{pmatrix} + t\begin{pmatrix}2\\1\\-1\end{pmatrix}$ are skew.(a) Find a vector perpendicular to both lines.(b) Hence find the shortest distance between $L_1$ and $L_2$.

(a) Show that $\cos 3\theta = 4\cos^3\theta – 3\cos\theta$.(b) Hence solve $4\cos^3\theta – 3\cos\theta = \tfrac{1}{2}$ for $0 \leq \theta \leq \pi$, giving exact answers.

Two planes have equations $\Pi_1: 2x – y + z = 4$ and $\Pi_2: x + y – 3z = 1$.(a) Find the acute angle between the planes.(b) Find a vector equation of the line of intersection of $\Pi_1$ and $\Pi_2$.

Write $\sin\theta – \sqrt{3}\cos\theta$ in the form $R\sin(\theta – \alpha)$, $R > 0$, $0 < \alpha < \tfrac{\pi}{2}$. Hence find the maximum value of $\dfrac{6}{4 + \sin\theta – \sqrt{3}\cos\theta}$ and the smallest positive value of $\theta$ at which it occurs.

A particle moves so that its position at time $t$ seconds is $\mathbf{r}(t) = \begin{pmatrix}3 + 2t\\1 – t\\4 + 2t\end{pmatrix}$ metres. A second particle has position $\mathbf{s}(t) = \begin{pmatrix}11 – 2t\\-3 + t\\10 – t\end{pmatrix}$.(a) Find the speed of the first particle.(b) Show that the particles collide, and find the time and position of the collision.(c) Find the angle between the two paths.