IB Maths AA HL 3.2 Geometry and trigonometry - AHL content Question Bank

Consider the two planes

Find the angle between $\pi_{1}$ and $\pi_{2}$, giving your answer correct to the nearest degree.

The following system of equations represents three planes in space.

Find the coordinates of the point of intersection of the three planes.

The points A and B have position vectors $\overrightarrow{\mathrm{OA}}=\left(\begin{array}{c}1 \\ 2 \\ -2\end{array}\right)$ and $\overrightarrow{\mathrm{OB}}=\left(\begin{array}{l}1 \\ 0 \\ 2\end{array}\right)$.

Let $\boldsymbol{a}=\left(\begin{array}{c}2 \\ k \\ -1\end{array}\right)$ and $\boldsymbol{b}=\left(\begin{array}{c}-3 \\ k+2 \\ k\end{array}\right), k \in \mathbb{R}$.
Given that a and b are perpendicular, find the possible values of k.

This question asks you to investigate lines normal to curves of the form $y=\frac{k^{2}}{x}$.
The curve H has equation $y=\frac{1}{x}$ where $x \in \mathbb{R}, x \neq 0$.

The acute angle between the vectors 3 i-4 j-5 k and 5 i-4 j+3 k is denoted by $\theta$. Find $\cos \theta$.

Find the coordinates of the point of intersection of the planes defined by the equations x+y+z=3, x-y+z=5 and x+y+2 z=6.

The following diagram shows a pyramid with vertex V and rectangular base O A B C.
Point B has coordinates ( 6,8,0 ), point C has coordinates ( 6,0,0 ) and point V has coordinates ( 3,4,9 ).

Three points in three-dimensional space have coordinates A(0,0,2), B(0,2,0) and C(3,1,0).

The points A and B are given by A(0,3,-6) and B(6,-5,11).
The plane $\Pi$ is defined by the equation 4 x-3 y+2 z=20.