Math topic中文

Geometrical meanings projections

Geometrical meanings of $\hat{\mathbf{a}} \cdot \hat{\mathbf{n}}$ and $\hat{\mathbf{a}} \times \hat{\mathbf{n}}$ , where $\hat{\mathbf{n}}$ is a unit vector.

Practice questions

1A drilling company plans to install a straight pipeline $AB$ through a mountain. Points $(x, y, z)$ are defined relative…
2With reference to the origin $O$ , the points $A$ , $P$ and $Q$ have position vectors $\mathbf{a}$ , $\mathbf{p}$ and $\mathbf{q}$ …
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 $\mathbf{r} = \lambda_1(\mathbf{i} + \mathbf{j} - \mathbf{k}) + \mu_1(2\mathbf{i} - \mathbf{j} + \mathbf{k}) \quad \text{and} \quad \mathbf{r} = \lambda_2(\mathbf{i} + 2\mathbf{j} + \mathbf{k}) + \mu_2(3\mathbf{i} + \mathbf{j} - \mathbf{k})$ …
5Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
6OABC is a rhombus. The points A and C are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$ …
7In triangle $PQR$ , the vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\vec{PQ}$ , $\vec{QR}$ …
8Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
9The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
10To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
11With reference to the origin $O$ , the position vectors of points $A$ , $B$ and $C$ are expressed as $\overrightarrow{OA}$ …
12Referred to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such…
13Relative to the origin O, points A and B have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ …
14The line $l$ has equation $1 - x = \frac{z - \alpha}{2}, y = 1$ , where $\alpha$ is a constant. Given that the point of i…
15Relative to the origin, the fixed point $A$ has position vector $\mathbf{a}$ . The line $l$ has equation $\mathbf{r} = s\mathbf{d}$ …
16The 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…
17The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a} = 4\mathbf{i} + 6p\mathbf{j} - 8\mathbf{k} \quad \text{and} \quad \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} + 4p\mathbf{k} \text{, where } p > 0.$ …
18Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
19With reference to the origin O, the point A has position vector $-\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$ . The plane $\pi$ …
20A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…
21The planes $\Pi_1$ and $\Pi_2$ have equations $\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 4$ and $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + \mathbf{k}) + \mu(4\mathbf{i} + \mathbf{j})$ …
22Given $a$ and $b$ are two non-zero and non-parallel vectors and $c = |b|a + |a|b$ , show that the length of projection of…
23The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. $C$ is the point on line $OB$ s…
24Relative to the origin $O$ , the points $A$ and $C$ have position vectors $\mathbf{a}$ and $2\mathbf{b}$ such that $\mathbf{a} = 5p\mathbf{i} - 2p\mathbf{j} + 4p\mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k},$ …
25The diagram shows a solid with a horizontal rectangular base $OABC$ in which the lengths of $OA$ and $OC$ are $12\text{ m}$ …
26Relative to the origin $O$ , three distinct fixed points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ …
27The molecular geometry is the three-dimensional arrangement of atoms in a molecule. Fig. 1 shows the molecular geometry…
28Relative to an origin $O$ , the points $A$ and $B$ have position vectors $4\mathbf{i} + 2\mathbf{j}$ and $3\mathbf{i} + \mathbf{k}$ …
29Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
30The diagram shows a unit circle with centre $O$ and diameter $AB$ . The point $C$ lies on the circumference of the circle…
31Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ …
32Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
33The diagram below shows a triangular base pyramid with vertices, $A$ , $B$ , $C$ and $D$ . With reference to the origin $O$ …
34The equations of three planes $p_1, p_2, p_3$ are $2x-4y+z=6,$ $x+y-z=6,$ $ax-2y-z=b,$ respectively, where $a$ and $b$ a…
35The points P, Q and R have position vectors p, q and r respectively, with respect to the origin O. The vector p is a uni…