3.18 Intersections of lines and planes questions for IB Maths AA

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

Hence find the area of triangle ABC.

Find the Cartesian equation of the plane $\mathrm{\Pi }$ that contains A, B and C.

The line $L$ passes through V and is perpendicular to $\mathrm{\Pi }$.

Write down a vector equation of $L$.

The line $L$ meets $\mathrm{\Pi }$ at the point F.

Find the coordinates of F.

Hence find the volume of the pyramid.

Find a vector that is normal to $\mathrm{\Pi }$.

Hence find the Cartesian equation of $\mathrm{\Pi }$.

Find the acute angle between $\mathrm{\Pi }$ and the plane with equation $z=0$.

Consider the case $a=2$, $b=3$ and $c=6$.

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

Hence show that, in this case, the equation of $\mathrm{\Pi }$ is $3x+2y+z=6$.

Now consider the general case.

Show that $\overrightarrow{\mathrm{A}\mathrm{B}}\times \overrightarrow{\mathrm{A}\mathrm{C}}=\left(\begin{array}{c}bc\\ ac\\ ab\end{array}\right)$.

Hence show that the equation of $\mathrm{\Pi }$ can be written as ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=1$.

Let $T$ be the area of triangle $\mathrm{A}\mathrm{B}\mathrm{C}$. Let $P$, $Q$ and $R$ be the areas of the right-angled triangles $\mathrm{O}\mathrm{B}\mathrm{C}$, $\mathrm{O}\mathrm{C}\mathrm{A}$ and $\mathrm{O}\mathrm{A}\mathrm{B}$ respectively.

For the case $a=2$, $b=3$ and $c=6$, find the value of $T$.

For the same case, find the values of $P$, $Q$ and $R$, and verify that $T^2=P^2+Q^2+R^2$.

Prove that $T^2=P^2+Q^2+R^2$ for all $a,b,c>0$.

Let $h$ be the perpendicular distance from $\mathrm{O}$ to $\mathrm{\Pi }$.

By considering the volume of the tetrahedron $\mathrm{O}\mathrm{A}\mathrm{B}\mathrm{C}$ in two different ways, show that

A plane meets the positive coordinate axes at $(k,0,0)$, $(0,k,0)$ and $(0,0,2k)$. The perpendicular distance from $\mathrm{O}$ to this plane is 4.

Find the value of $k$.

Let $\alpha$ be the acute angle between $\mathrm{\Pi }$ and the plane $x=0$, which contains triangle $\mathrm{O}\mathrm{B}\mathrm{C}$.

Show that $\cos \alpha ={\displaystyle \frac{P}{T}}$.

Let $\beta$ and $\gamma$ be the acute angles between $\mathrm{\Pi }$ and the planes $y=0$ and $z=0$ respectively.

Hence show that ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$.

Find $\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right)\times \left(\begin{array}{c}2\\ -1\\ 1\end{array}\right)$.

Verify that the point $\mathrm{P}(2,1,1)$ lies on both ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.

The planes ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ meet in the line $L$. Hence find a vector equation for $L$.

The plane ${\mathrm{\Pi }}_3$ contains the point $\mathrm{Q}(0,-3,-4)$ and is perpendicular to $L$.

Find the Cartesian equation of ${\mathrm{\Pi }}_3$.

The line $L$ meets ${\mathrm{\Pi }}_3$ at the point R.

Find the coordinates of R.

Find the shortest distance from Q to $L$.

The point ${\mathrm{Q}}^{\mathrm{\prime }}$ is the reflection of Q in the line $L$.

Find the coordinates of ${\mathrm{Q}}^{\mathrm{\prime }}$.

Show that the Cartesian equation of $\mathrm{\Pi }$ is $2x+3y=7$.

The line $L$ has vector equation $\mathbf{r}=\lambda \left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$, where $\lambda \in \mathbb{R}$.

Find the acute angle between $L$ and $\mathrm{\Pi }$.

Consider the plane $\mathrm{\Pi }$ with equation $2x-y+2z=7$ and the point $\mathrm{P}(5,1,4)$.

Write down a vector equation of the line through $\mathrm{P}$ that is perpendicular to $\mathrm{\Pi }$.

This line meets $\mathrm{\Pi }$ at the point $\mathrm{F}$. Find the coordinates of $\mathrm{F}$.

Hence find the perpendicular distance from $\mathrm{P}$ to $\mathrm{\Pi }$.

Now consider the general plane $ax+by+cz=d$ and a point $\mathrm{P}(x_0,y_0,z_0)$ that does not lie on the plane.

Show that the line through $\mathrm{P}$ perpendicular to the plane meets the plane at the point with parameter

Hence show that the perpendicular distance from $\mathrm{P}$ to the plane is

Use this formula to verify your answer to part (a)(iii).

Show that the distance between the parallel planes $ax+by+cz=d_1$ and $ax+by+cz=d_2$ is ${\displaystyle \frac{\mid d_1-d_2\mid }{\sqrt{a^2+b^2+c^2}}}$.

Consider the planes ${\mathrm{\Pi }}_1$: $2x-y+2z=3$ and ${\mathrm{\Pi }}_2$: $6x+2y-3z=4$.

Show that the set of points that are equidistant from ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ consists of the two planes

Show that these two planes are perpendicular.

Explain why both of these planes contain the line of intersection of ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.

Find the coordinates of the points on the $z$-axis that are equidistant from ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.

Suggest a geometrical description of the two planes found in part (d)(i), in relation to ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.