Coincident, Parallel, intersecting, skew lines: notes and practice questions

The path of a drone, $D_{1}$, can be modelled by the vector equation $r = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$. A communication beacon is located at point B with coordinates $(k, 3, 11)$. The drone passes through the location of the beacon.

(a) Find the value of $k$.

A second drone, $D_{2}$, starts at the point $(0, 0, 1)$ and travels on a path parallel to $D_{1}$.

(b) Write down a vector equation for the path of $D_{2}$.

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.

Consider the line $L_1$ with vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$.

(a) (i) Find a Cartesian equation for the line $L_1$.

(ii) Show that the point $P(4, 8, -4)$ lies on $L_1$.

Consider a second line $L_2$ with direction vector $\begin{pmatrix} p \\ 1 \\ 3 \end{pmatrix}$. The acute angle between $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(b) Find the possible values of $p$.

Let a third line, $L_3$, be defined by the vector equation $\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} q \\ 2 \\ -4 \end{pmatrix}$. The lines $L_1$ and $L_3$ intersect at a point B.

(c) Find the value of $q$ and the coordinates of B.

(a) The line $L_1$ passes through the point Q(2, 0, 5) and has a direction vector $\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$.

Write down a vector equation for $L_1$.

(b) A second line, $L_2$, passes through the points C(3, 1, 0) and D(4, 3, -2).

Find a vector equation for $L_2$.

(c) Show that $L_1$ and $L_2$ are skew.

(d) Find $\vec{\text{QM}} \cdot \vec{\text{CD}}$ in terms of $\mu$, where M is a general point on $L_2$.

(e) Hence, find the coordinates of the point M on $L_2$ that is closest to Q.

(f) The origin is denoted by O(0, 0, 0). Find the equation of the plane that contains the points O, Q and the point M found in part (e). Give your answer in the form $ax + by + cz = d$, where $a, b, c, d \in \mathbb{Z}$.

Points P and Q have position vectors $\vec{p}$ and $\vec{q}$ respectively, relative to an origin O. Let M be the midpoint of the line segment [PQ].

Show that the position vector of M is $\vec{m} = \frac{1}{2}(\vec{p} + \vec{q})$.

A triangle has vertices P(1, 0, 2), Q(3, 4, -2), and R(5, 2, 6).

Let L, M and N be the midpoints of the sides [PQ], [QR] and [RP] respectively. Find the position vectors of L, M and N.

The centroid G of the triangle PQR has position vector $\vec{g} = \frac{1}{3}(\vec{p} + \vec{q} + \vec{r})$.

Find the coordinates of G.

Show that the points P, G, and M are collinear, where M is the midpoint of [QR].

Hence, find the ratio PG:GM.

Let P(1, 0, 1), Q(1, 2, 0), and R(k+1, 1, -1) be three points in $\mathbb{R}^3$, where k > 0.

Let $Π$ be the plane containing the points P, Q, and R.

(a) Find a Cartesian equation for the plane $Π$ in terms of k.

(b) Let N be the midpoint of the line segment [PR]. A line L passes through N and is perpendicular to the plane $Π$. Find a vector equation for the line L in terms of k.

(c) Let $L'$ be the line defined by the equations $y=x, z=1$. Show that the line L does not intersect the line $L'$ for any k > 0.