Math topic中文

Equations of lines and planes

Vector and cartesian equations of lines and planes.

Practice questions

1The lines $l_1$ and $l_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ -6 \end{pmatrix}, \lambda \in \mathbb{R}$ …
2A drilling company plans to install a straight pipeline $AB$ through a mountain. Points $(x, y, z)$ are defined relative…
3A plane $\Pi_1$ has equation $x + y + z = 3$ . A line passes through the points $P$ and $Q$ with position vectors $\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ …
4The planes $\Pi_1$ and $\Pi_2$ have vector equations $\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$ …
5A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
6The equations of three planes $p_1$ , $p_2$ and $p_3$ are $\begin{aligned} 4x + 6y - z &= 3 \\ 2x + 5y - 2z &= 1 \\ \alpha x - 3y + 4z &= \beta \end{aligned}$ …
7The diagram below shows a tetrahedron $ABCD$ . The equation of the plane $ABD$ is $4x + y + 2z = 16$ . Given that $A$ is o…
8The points $A$ and $B$ have coordinates $(0, 2, 0)$ and $(1, 0, 2)$ respectively, and the plane $\pi_1$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} = 4$ …
9A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a t…
10The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
11The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
12The position vectors of the points $A$ and $B$ relative to the origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively…
13Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ …
14Relative to the origin $O$ , the points $A$ and $B$ have non-zero and non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ …
15Planes $p$ and $q$ are perpendicular. Plane $p$ has equation $2x - z = 10$ . Plane $q$ contains the line $l$ with equatio…
16The points A, B and C have position vectors $2i,\;3j$ and $4k$ respectively. Find a vector which is perpendicular to the…
17In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i,…
18The two planes $p_{1}$ and $p_{2}$ , given by the equations $2x+y=1$ and $8x+ay+z=4$ respectively, meet at a line $L$ whi…
19Relative to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
20Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
21The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
22In a computer game, a slope can be modelled as a plane $p$ containing three points, $A(1, 0, -3)$ , $B(1, 4, -15)$ and $C(2, 3, -14)$ …
23The plane $p$ passes through the points $A$ , $B$ and $C$ with coordinates $(0, 1, 1)$ , $(2, -1, 4)$ and $(-2, -1, 0)$ re…
24The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
25Referred to the origin $O$ , the points $P$ and $Q$ have position vectors $4\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $8\mathbf{i} + 8\mathbf{j} - \mathbf{k}$ …
26The points A, B and C have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ respectively. The points A and B…
27The point A has coordinates (1, -2, 4) and the plane $\pi_1$ has equation $2x - y + 2z = 5.$ Find the exact shortest dis…
28With respect to the origin $O$ , the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respect…
29The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
30To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
31A tetrahedron $OABC$ has four triangular faces with a vertex $O$ at the origin. The coordinates of $A$ , $B$ and $C$ are…
32The lines $l_1$ and $l_2$ have equations $\frac{x-1}{3} = \frac{y-2}{a}, z = 1 \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \lambda \in \mathbb{R}$ …
33The planes p and q have equations $r = (2 + \lambda - 2\mu)\,\mathbf{i} + (-3\lambda + \mu)\,\mathbf{j} + (3 - \lambda + 2\mu)\,\mathbf{k}$ …
34The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
35Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
36An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. The…
37The planes $\Pi_1$ and $\Pi_2$ , which meet in the line $l$ , have vector equations $\mathbf{r} = 2\mathbf{i} + 4\mathbf{j} + 6\mathbf{k} + \theta_1(2\mathbf{i} + 3\mathbf{k}) + \phi_1(-4\mathbf{j} + 5\mathbf{k}),$ …
38The diagram shows a pyramid DOABC. Taking unit vectors i, j, k as shown, the position vectors of A, B, C, D are given by…
39The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
40The line $l_1$ has the vector equation $\mathbf{r}=\begin{pmatrix}a\\1\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\1\\b\end{pmatrix},\ \lambda\in\mathbb{R},$ …
41The equations of planes $p_1$ and $p_2$ are respectively given by $\mathbf{r} = s \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}, \quad s, t \in \mathbb{R} \quad \text{and} \quad \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 3 \\ \beta \end{pmatrix} = 7,$ …