Math topic中文

Position displacement direction vectors

Position vectors, displacement vectors and direction vectors.

Practice questions

1A drilling company plans to install a straight pipeline $AB$ through a mountain. Points $(x, y, z)$ are defined relative…
2A plane $\Pi_1$ has equation $x + y + z = 3$ . A line passes through the points $P$ and $Q$ with position vectors $\mathbf{i} + \mathbf{j} + 2\mathbf{k}$ …
3A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
4The points $A$ and $B$ have coordinates $(0, 2, 0)$ and $(1, 0, 2)$ respectively, and the plane $\pi_1$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} = 4$ …
5A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a t…
6The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
7In the triangle shown below, one vertex is origin O, and the two other vertices are A and B where $\overrightarrow{OA}=\mathbf{a}$ …
8Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
9The equations of planes $p_1$ and $p_2$ are given by $p_1:\;\mathbf{r}\cdot \begin{pmatrix}\alpha\\7\\2\end{pmatrix} = \alpha + 20,\; \alpha \in \mathbb{R},$ …
10The position vectors of the points $A$ and $B$ relative to the origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively…
11Referred to the origin O, the position vectors of the points A and B are $i - j + 2k\quad\text{and}\quad 2i + 4j + k$ re…
12Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ …
13Relative to the origin $O$ , the points $A$ and $B$ have non-zero and non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ …
14Referred to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
15Three non-zero vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ are such that $3\mathbf{p} \times \mathbf{q} = \mathbf{r} \times \mathbf{p}$ …
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,…
18OABC is a rhombus. The points A and C are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$ …
19The 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…
20Relative to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
21Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
22The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
23With respect to an origin O, the points A, B, C, D have position vectors $2i - j + k,\quad i - 2k,\quad -i + 3j + 2k,\quad -i + j + 4k,$ …
24With reference to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectivel…
25In a computer game, a slope can be modelled as a plane $p$ containing three points, $A(1, 0, -3)$ , $B(1, 4, -15)$ and $C(2, 3, -14)$ …
26Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
27The plane $p$ passes through the points $A$ , $B$ and $C$ with coordinates $(0, 1, 1)$ , $(2, -1, 4)$ and $(-2, -1, 0)$ re…
28The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
29Referred to the origin $O$ , the points $P$ and $Q$ have position vectors $4\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $8\mathbf{i} + 8\mathbf{j} - \mathbf{k}$ …
30The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
31To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
32With reference to the origin $O$ , the position vectors of points $A$ , $B$ and $C$ are expressed as $\overrightarrow{OA}$ …
33A tetrahedron $OABC$ has four triangular faces with a vertex $O$ at the origin. The coordinates of $A$ , $B$ and $C$ are…
34The lines $l_1$ and $l_2$ have equations $\frac{x-1}{3} = \frac{y-2}{a}, z = 1 \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}, \lambda \in \mathbb{R}$ …
35The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
36Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
37An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. The…
38Relative to the origin O, points A and B have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ …
39The planes $\Pi_1$ and $\Pi_2$ , which meet in the line $l$ , have vector equations $\mathbf{r} = 2\mathbf{i} + 4\mathbf{j} + 6\mathbf{k} + \theta_1(2\mathbf{i} + 3\mathbf{k}) + \phi_1(-4\mathbf{j} + 5\mathbf{k}),$ …
40The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
41Relative to the origin $O$ , the position vectors of points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively, w…
42Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 2\mathbf{i} - \mathbf{j}$ …
43Relative to the origin, the fixed point $A$ has position vector $\mathbf{a}$ . The line $l$ has equation $\mathbf{r} = s\mathbf{d}$ …
44The 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…
45With reference to the origin O, the points N, P and Q have position vectors $\mathbf{n}$ , $\mathbf{p}$ and $\mathbf{q}$ …
46Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
47Referred to the origin O, let P, Q and R be distinct points with position vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ …
48A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horiz…
49Two 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…
50The 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…
51Referred to an origin $O$ , the position vectors of three points $A$ , $B$ and $C$ are $3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$ …
52The 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…
536 equilateral triangles each with sides 5 units are arranged to form a regular hexagon as shown in the diagram below. By…
54Referred to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
55With reference to the origin $O$ , the points $A$ , $B$ and $C$ are such that $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ …
56With reference to the origin O, the point A has position vector $-\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$ . The plane $\pi$ …
57Referred to the origin O, let P, Q and R be distinct points with position vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ …
58The points $P$ and $Q$ have position vectors $3\mathbf{i}+4\mathbf{j}-\mathbf{k}$ and $5\mathbf{i}+7\mathbf{j}+6\mathbf{k}$ …
59Relative to the origin $O$ , two fixed points, $A$ and $B$ , have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectiv…
60A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…