Intersection and angles between lines & planes: notes and practice questions

The paths of two submarines, A and B, are described by the vector equations below, where $s, t \in \mathbb{R}$ are time parameters in hours, and the coordinates are in kilometres.

$L_A: \boldsymbol{r} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$

$L_B: \boldsymbol{r} = \begin{pmatrix} 11 \\ 2 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$

(a) Show that the paths of the two submarines do not cross.

(b) Find the shortest distance between the paths of the two submarines.

The plane $\Pi_1$ has equation $x - 2y + 3z = 1$.

(a) Show that the point $A(2, 2, 1)$ lies on the plane $\Pi_1$.

The plane $\Pi_2$ is given by $ax + (a^2 - 15)y + bz = d$, where $a, b, d \in \mathbb{R}$ and $b \neq 0$.

(b) In the case where $b = 5$, $\Pi_2$ is perpendicular to $\Pi_1$ and point A lies on $\Pi_2$. Given that $a > 0$, find the value of $a$ and the value of $d$.

For parts (c), (d) and (e) it is now given that $\Pi_2$ is parallel to $\Pi_1$.

(c) Given that $a=3$, determine the value of $b$.

It is also given that $d = 31$.

The line through A that is perpendicular to $\Pi_1$ meets $\Pi_2$ at the point B.

(d) (i) Find the coordinates of B.

(ii) Hence, find the perpendicular distance between $\Pi_1$ and $\Pi_2$.

(e) Find the equation of a third parallel plane $\Pi_3$ which is also a perpendicular distance of $\frac{2\sqrt{14}}{3}$ from $\Pi_1$.

A team of architects is designing a new building and needs to define a support beam's orientation. The beam must be perpendicular to two existing structural walls, Wall A and Wall B. The equations of the planes representing these walls are given by:

Wall A ($\Pi_A$): $x + 2y - z = 7$

Wall B ($\Pi_B$): $2x - y + 3z = 1$

Find a Cartesian equation of the plane ($\Pi_C$) that represents the orientation of the support beam, given that it passes through the origin (0, 0, 0).

Find the coordinates of the point where Wall A, Wall B, and the support beam's plane ($\Pi_C$) intersect.

Three points $D(1, 2, 0)$, $E(3, 0, -1)$ and $F(0, 1, 4)$ lie on the plane $\Pi_1$.

Find the vector $\vec{DE}$ and the vector $\vec{DF}$.

Hence find the equation of $\Pi_1$, expressing your answer in the form $ax + by + cz = d$, where $a, b, c, d \in \mathbb{Z}$.

Plane $\Pi_2$ has equation $x + y - z = 2$.

The line $L$ is the intersection of $\Pi_1$ and $\Pi_2$. Verify that the vector equation of $L$ can be written as $\mathbf{r} = \begin{pmatrix} -1 \\ 4 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -11 \\ 13 \\ 2 \end{pmatrix}$.

The plane $\Pi_3$ is given by $x + 2y - z = 5$. The line $L$ and the plane $\Pi_3$ intersect at the point $P$.

Show that at the point $P$, $\lambda=-\frac{1}{13}$.

Hence find the coordinates of $P$.

The point $Q(-1, 4, 1)$ lies on $L$.

Find the reflection of the point $Q$ in the plane $\Pi_3$.

Hence find the vector equation of the line formed when $L$ is reflected in the plane $\Pi_3$.

The path of a particle is modelled by the line $L$ with vector equation $r = \begin{pmatrix} 2 \\ 1 \\ 8 \end{pmatrix} + t \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}$. The particle collides with a flat surface modelled by the plane $\pi$ with equation $x - 2y + 2z = 5$. The point of collision is $Q$. Find the coordinates of $Q$.

The particle starts its path at the point $A(2, 1, 8)$. Find the shortest distance from the starting point $A$ to the surface $\pi$.

Consider the points given by the coordinates $P(2, 1, 0)$, $Q(0, 3, 1)$, $R(1, 0, 4)$.

Find the vector $\vec{PQ} \times \vec{PR}$.

Hence, find the exact area of triangle PQR.

Show that the Cartesian equation of the plane $\Pi_1$, which contains the triangle PQR, is $9x + 7y + 4z = 25$.

A second plane is given by the equation $\Pi_2: x + y - z = 1$. Find a vector equation for the line of intersection of the planes $\Pi_1$ and $\Pi_2$.

A satellite dish is positioned at a ground control station $A(3, 1, 4)$. The dish is designed to track a celestial object whose path can be modelled by a line $L_1$ with vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$, where $t \in \mathbb{R}$.

The plane $\Pi_1$ of the satellite dish contains the line $L_1$ and passes through the ground control station $A$.

Show that the Cartesian equation of the plane $\Pi_1$ is $x + 2y = 5$.

Consider three large display screens in a museum, represented by the planes:

$\Pi_1 : x + 2y = 5$

$\Pi_2 : 3x + ay - z = 7$

$\Pi_3 : 2x - y + z = k$

where $a, k \in \mathbb{Q}$.

For a special holographic effect, the three planes must intersect along a single line.

Find the value of $a$ and the value of $k$.

A laser beam is emitted from a source at point $A(3, -1, 5)$. The beam reflects off a flat mirror which lies on the plane $\pi: x - 2y + 2z = 6$. The reflected beam appears to originate from a virtual source at point $B$, where $B$ is the reflection of $A$ in the plane $\pi$.

Determine the coordinates of $B$.

Find the exact distance between the laser source $A$ and the virtual source $B$.

Consider the three points P(4,0,1), Q(0,−3,1), and R(2,2,−5) lie on a plane $\Pi_{1}.$

$a$ Find the vector $\overrightarrow{PQ}$ and the vector $\overrightarrow{PR}$.

$b$ Find the cartesian equation of plane $\Pi_{1}$.

$c$ Find the equation of the line L that passes through the point S(-8,1,23) and perpendicular to $\Pi_{1}$ .

$d$ Find the coordinates of the point of intersection between line L and plane $\Pi_{1}$.

A structural engineer is designing a framework and needs to define the orientation of certain surfaces. Consider two existing planar surfaces, $\Pi_1$ and $\Pi_2$, with the following Cartesian equations:

$\Pi_1\text{: } 2x + y - z = 5$

$\Pi_2\text{: } x - 3y + 2z = -1$

Find a Cartesian equation of a third planar surface, $\Pi_3$, which is perpendicular to both $\Pi_1$ and $\Pi_2$, and passes through the point $(1, 2, 3)$.

Determine the coordinates of the point where $\Pi_1$, $\Pi_2$, and $\Pi_3$ intersect.

A plane $\Pi$ has the Cartesian equation $x + 3y - 2z = 5$. A point B has coordinates $(1, 0, 4)$.

(a) Find the vector equation of the line $L$ that passes through the point B and is perpendicular to the plane $\Pi$.

(b) Find the coordinates of the point of intersection, N, of the line $L$ and the plane $\Pi$. Hence, find the exact distance between the point B and the plane $\Pi$.

(c) The point P has coordinates $(x_0, y_0, z_0)$.

Show that the distance between the point P and the plane $\Pi$ is given by

$\frac{|x_0 + 3y_0 - 2z_0 - 5|}{\sqrt{14}}$

A laser beam is modelled by the line $L$ with equation $\vec{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}$. The beam strikes a flat mirror surface, which lies on the plane $\Pi$ with equation $2x - y + 3z = 5$.

(a) Find the coordinates of the point where the laser beam hits the mirror.

(b) Determine the acute angle between the laser beam and the mirror surface.

(c) Find the vector equation of the reflected laser beam.

A drone is programmed to follow a straight flight path $L$. The path is described by the Cartesian equation $L: \frac{x+3}{4} = \frac{y-1}{2} = z+2$.

Find the vector equation of the drone's flight path $L$, expressing your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\lambda \in \mathbb{R}$.

A ground control station is located at the origin $O(0,0,0)$.

Determine the minimum distance from the ground control station to the drone's flight path $L$.

A security laser grid is set up, forming a plane $\Pi$ with the equation $\Pi: x+y-6z=10$.

Verify that the drone's flight path $L$ lies entirely within the security laser grid $\Pi$.

A second drone is launched from a point $P(5, 2, -1)$. This drone's flight path, $M$, is parallel to the security laser grid $\Pi$ and is designed to intersect the $y$-axis.

Find the vector equation of the second drone's flight path $M$, expressing your answer in the form $\mathbf{s} = \mathbf{c} + \mu\mathbf{d}$, where $\mu \in \mathbb{R}$.