11c. Lines and planes

Work in progress

The following questions illustrate the key concepts needed in this topic. See if you can answer them.
Click on the title for detailed discussion of how to tackle them.
Click on or hover over the question to show the answer.

Consider the points $A(3,-1,6), B(1,-1,-4)$ and $C(5,3,0)$.

Find the equation of line $l_1$ which contains points $A$ and $C$. $\mathbf{r}=(3\mathbf{i}-\mathbf{j}+6\mathbf{k}) + \lambda (\mathbf{i}+2\mathbf{j}-3\mathbf{k}), \lambda \in \mathbb{R}$.

Find, in cartesian form, the equation of the line that contains point $B$ and is parallel to $l_1$. $x-1 = \frac{y+1}{2}=\frac{z+4}{-3}$.

Consider the points $A(3,-1,6), C(5,3,0), D(5,-2,5)$ and the lines
$l_1: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ l_2: \mathbf{r} =\begin{pmatrix} 1 \\ -1 \\ -4 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -1 \\ -1 \end{pmatrix}, \mu \in \mathbb{R}$.

Find, in vector form, the equation of the plane containing points $A,C$ and $D$. $\mathbf{r} = (3\mathbf{i}-\mathbf{j}+6\mathbf{k})+\lambda(\mathbf{i}+2\mathbf{j}-3\mathbf{k})+\mu(2\mathbf{i}-\mathbf{j}-\mathbf{k}), \mu \in \mathbb{R}$

Find, in scalar product form, the equation of the plane which contains line $l_1$ and the point $D$. $\mathbf{r} \cdot (\mathbf{i}+\mathbf{j}+\mathbf{k}) = 8$

Find, in cartesian form, the equation of the plane which contains line $l_1$ and is parallel to line $l_2$. $x+y+z=8$.

The lines $l$ and $l_3$ and the planes $\pi_1$ and $\pi_2$ have equations
$ l_1: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ l_3: \mathbf{r} = \begin{pmatrix} 5 \\ 3 \\ 0 \end{pmatrix} + \nu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}, \nu \in \mathbb{R} \\ \pi_1: x+y+z = 8 \\ \pi_2: 2x-3y+5z=-2$.

Find the acute angle between $l_1$ and $l_3$. $49.1^\circ$

Find the angle between $l_1$ and $\pi_2$ $55.5^\circ$

Find the cosine of the angle between $\pi_1$ and $\pi_2$. $\frac{4}{\sqrt{114}}$

Consider the lines $l$ and $m$ and the planes $\pi_1$ and $\pi_2$ with equations:
$ l: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ m: \mathbf{r} = \begin{pmatrix} 5 \\ 4 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}, \mu \in \mathbb{R} \\ \pi_1: x+y+z = 8 \\ \pi_2: 2x-3y+5z=-2$.

Find the coordinates of the point of intersection between $l$ and $m$. $ (5,3,0) $

Find the position vector of the point of intersection between $l$ and $\pi_2$. $\frac{1}{19}(98\mathbf{i}+63\mathbf{j}-9\mathbf{k})$

Find the equation of the line of intersection between $\pi_1$ and $\pi_2$. $\mathbf{r}=(\frac{22}{5}\mathbf{i}+\frac{18}{5}\mathbf{j}) + \lambda (\mathbf{i}+2\mathbf{j}+3\mathbf{k})$

The point $B$ has coordinates $(1,-1,4)$. The equations of the line $l$ and the plane $\Pi$ are:
$ l: \mathbf{r}=(3\mathbf{i}-\mathbf{j}+6\mathbf{k}) + \lambda (\mathbf{i}+2\mathbf{j}-3\mathbf{k}), \lambda \in \mathbb{R} \\ \Pi: 2x-3y+5z = -2$.

Find the perpendicular distance from $B$ to line $l$. $4\sqrt{3}$

Find the shortest distance between $B$ and $\Pi$. $\sqrt{\frac{69}{38}}$

The point $B$ has coordinates $(1,-1,4)$. The equations of the line $l$ and the plane $\pi$ are:
$ l: \mathbf{r}=(3\mathbf{i}-\mathbf{j}+6\mathbf{k}) + \lambda (\mathbf{i}+2\mathbf{j}-3\mathbf{k}), \lambda \in \mathbb{R} \\ \pi: x+y+z = 8$.

Find the coordinates of the point on line $l$ that is closest to $B$. $(5,3,0)$

The point $F$ is the foot of the perpendicular from $B$ to $\pi$. Find the position vector of $F$. $5\mathbf{i}+3\mathbf{j}$

Find the coordinates of the point $B'$ that is obtained by reflecting $B$ in the line $l$. $(9,7,4)$

Consider the lines $l, m_1, m_2$ and $m_3$ and the planes $\Pi_1$ and $\Pi_2$ with equations: $ l: \mathbf{r} =\begin{pmatrix} 3 \\ -1 \\ 6 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \lambda \in \mathbb{R} \\ m_1: \mathbf{r} =\begin{pmatrix} 7 \\ 6 \\ -5 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 3 \\ -5 \end{pmatrix}, \mu \in \mathbb{R} \\ m_2: \mathbf{r} =\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix} + \nu \begin{pmatrix} 2 \\ 3 \\ -5 \end{pmatrix}, \nu \in \mathbb{R} \\ m_3: \mathbf{r} =\begin{pmatrix} 4 \\ 0 \\ 7 \end{pmatrix} + \omega \begin{pmatrix} -2 \\ 4 \\ 6 \end{pmatrix}, \omega \in \mathbb{R} \\ \Pi_1: ax+by = 4 \\ \Pi_2: cx+y-z=d$.

State a pair of lines that are parallel. $l \textrm{ and } m_3$

Determine if $l$ and $m_1$ intersect. $\textrm{They intersect at } (5,3,0)$

Show that $l$ and $m_2$ are skew lines.

Given that $m_2$ lies on $\Pi$, find $a$ and $b$. $a=1, b =- \frac{2}{3}$

Given instead that $m_2$ intersects $\Pi$ at exactly one point, what can we say about $a$ and $b$? $2a+3b \neq 0$

If $l$ and $\Pi_2$ have no common points, find $c$ and write down what value $d$ cannot take. $c=-5, d \neq 20$