Math topic中文

Angles

Angle between two lines, between a line and a plane, or between two 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…
3With reference to the origin $O$ , the points $A$ , $P$ and $Q$ have position vectors $\mathbf{a}$ , $\mathbf{p}$ and $\mathbf{q}$ …
4The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
5The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
6The points A, B and C have position vectors $2i,\;3j$ and $4k$ respectively. Find a vector which is perpendicular to the…
7In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V. Perpendicular unit vectors i,…
8The 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…
9The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
10With respect to an origin O, the points A, B, C, D have position vectors $2i - j + k,\quad i - 2k,\quad -i + 3j + 2k,\quad -i + j + 4k,$ …
11The plane $p$ passes through the points $A$ , $B$ and $C$ with coordinates $(0, 1, 1)$ , $(2, -1, 4)$ and $(-2, -1, 0)$ re…
12The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
13Referred to the origin $O$ , the points $P$ and $Q$ have position vectors $4\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $8\mathbf{i} + 8\mathbf{j} - \mathbf{k}$ …
14The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
15A tetrahedron $OABC$ has four triangular faces with a vertex $O$ at the origin. The coordinates of $A$ , $B$ and $C$ are…
16The planes p and q have equations $r = (2 + \lambda - 2\mu)\,\mathbf{i} + (-3\lambda + \mu)\,\mathbf{j} + (3 - \lambda + 2\mu)\,\mathbf{k}$ …
17An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. The…
18The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
19The line $l_1$ has the vector equation $\mathbf{r}=\begin{pmatrix}a\\1\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\1\\b\end{pmatrix},\ \lambda\in\mathbb{R},$ …
20The plane $\Pi_1$ has equation $\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \theta(2\mathbf{j} - \mathbf{k}) + \phi(3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}).$ …
21The straight line $l$ has equation $\mathbf{r} = i + 6j - 3k + s(i - 2j + 2k).$ The plane $p$ has equation $(\mathbf{r} - 3i).(2i - 3j + 6k) = 0.$ …
22The plane $\Pi_1$ contains the points $A(1,4,2),\ B(1,0,5)$ and $C(0,8,-1)$ . Find a cartesian equation of $\Pi_1$ . Find…
23A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horiz…
24Two lines, $l_1$ and $l_2$ , have equations $\frac{x - 1}{2} = -z - 1, y = 2$ and $x + 3 = 4 - y = z + 2$ respectively. D…
25The plane $p_1$ has equation $x - y + z = 0$ and the plane $p_2$ has equation $x + y + z = 2$ . Find the acute angle betw…
26The line l has equation $\frac{x-1}{-2}=y=\frac{z+7}{4}$ , and the plane p has equation $x-z=2$ . Find the acute angle bet…
27With reference to the origin O, the point A has position vector $-\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$ . The plane $\pi$ …
28A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…
29The planes $\pi_1$ and $\pi_2$ , which meet in the line $l_1$ , have equations $\pi_1 : \mathbf{r} \cdot \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = 25 \text{ and } \pi_2 : x + ky - 2z = -15,$ …
30The planes $\Pi_1$ and $\Pi_2$ have equations $\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 4$ and $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + \mathbf{k}) + \mu(4\mathbf{i} + \mathbf{j})$ …
31A tetrahedron has four vertices O, A, B and C where O is the origin. It is given that the coordinates of B are (1, 4, 6)…
32The diagram above shows a toy house with horizontal rectangular base $ABCD$ where $AB = 30\text{ cm}$ and $BC = 20\text{ cm}$ …
33A frigate is stationed at position $F(1, 2, 0)$ . Two submarines $S_1$ and $S_2$ are under the sea surface. Submarine $S_1$ …
34The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).\n\nThe points A and B on$ …
35The diagram shows a solid with a horizontal rectangular base $OABC$ in which the lengths of $OA$ and $OC$ are $12\text{ m}$ …
36The line $l_1$ and the plane $\pi_1$ have equations $\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}, \lambda \in \mathbb{R}$ …
37With reference to the point $O$ as the origin and the $x$ - $y$ plane as a horizontal plane, the pyramid $OPQRV$ has a par…
38The diagram above shows a cuboid $OABCDEFG$ with horizontal base $OABC$ where $OA = 6\text{ cm}$ and $AB = 4\text{ cm}$ .…
39The line $l$ contains the point $P$ with coordinates $(1, -5, -5)$ and is parallel to $3\mathbf{i} + \mathbf{j} - 6\mathbf{k}$ …
40Show that the lines given by $\mathbf{r} = (5i + 2j + 4k) + \lambda(i + 3j + k)$ and $\mathbf{r} = (3i + j + k) + \mu(4i + 7j + 5k)$ …
41For positive real numbers $a$ , and $b$ , the lines $l_1$ and $l_2$ have equations given by $l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 0 \\ 3 \end{pmatrix} \text{ for } \lambda \in \mathbb{R} \quad \text{and} \quad l_2: \mathbf{r} = \begin{pmatrix} a \\ 9 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 7 \\ b \end{pmatrix} \text{ for } \mu \in \mathbb{R}.$ …
42The plane $\pi$ passes through the points with coordinates $(-1,0,1)$ , $(2,1,-1)$ and $(1,-3,2)$ . The line $l$ passes th…
43The plane $p_1$ has equation $x - y + z = 0$ and the plane $p_2$ has equation $x + y + z = 2$ . With reference to the ori…
44The diagram below shows a triangular base pyramid with vertices, $A$ , $B$ , $C$ and $D$ . With reference to the origin $O$ …
45With reference to the point $O$ as the origin and the x-y plane as a horizontal plane, the pyramid $OPQRV$ has a paralle…
46The line l passes through the points A and B with coordinates (1, 2, 4) and (-2, 3, 1) respectively. The plane p has equ…
47For an upcoming motor-car race, a spectators' gallery is to be set up near the racing track. As part of the preparations…
48A group of boys want to set up a camping tent. They lay down a rectangular tarp $OABC$ on the horizontal ground with $OA = 3\text{ m}$ …
49The equations of four planes $p_{1}$ , $p_{2}$ , $p_{3}$ and $p_{4}$ are $x+y+z=2,\\-x+ay+z=1,\\2x+2y+2z=b,\\cx+2y+3z=d,$ …