Math topic中文

Angle between two vectors

Angle between two vectors.

Practice questions

1The lines $l_1$ and $l_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ -6 \end{pmatrix}, \lambda \in \mathbb{R}$ …
2The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\mathbf{i}+(\sin\theta)\mathbf{j}+(\cos\theta)\mathbf{k}\quad$ …
3The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
4Referred to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
5Three non-zero vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ are such that $3\mathbf{p} \times \mathbf{q} = \mathbf{r} \times \mathbf{p}$ …
6Relative to the origin O, the position vectors of two points A and B are a and b respectively. The length of a is k unit…
7In triangle $PQR$ , the vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\vec{PQ}$ , $\vec{QR}$ …
8It is given that $a$ , $b$ and $c$ are non-zero vectors. If $|a+b| = |a-b|$ , show that the two vectors $a$ and $b$ are pe…
9Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
10The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=\mathbf{i}+(\sin\theta)\mathbf{j}+(\cos\theta)\mathbf{k}$ …
11With respect to an origin O, the points A, B, C, D have position vectors $2i - j + k,\quad i - 2k,\quad -i + 3j + 2k,\quad -i + j + 4k,$ …
12Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ …
13The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).$ …
14Referred to the origin $O$ , the points $P$ and $Q$ have position vectors $4\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$ and $8\mathbf{i} + 8\mathbf{j} - \mathbf{k}$ …
15The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
16Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
17Relative to the origin O, points A and B have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ …
18The diagram shows a pyramid DOABC. Taking unit vectors i, j, k as shown, the position vectors of A, B, C, D are given by…
19Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 2\mathbf{i} - \mathbf{j}$ …
20Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
21A designer wants to construct an inclined roof that is attached to a vertical rectangular wall, positioned above a horiz…
22Two 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…
23The 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…
24The above diagram shows a circle with radius $r$ units and centre $O$ . The points $A$ and $B$ on the circle are such tha…
25Referred to an origin $O$ , the position vectors of three points $A$ , $B$ and $C$ are $3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$ …
266 equilateral triangles each with sides 5 units are arranged to form a regular hexagon as shown in the diagram below. By…
27Referred to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
28The angle between the vector $\lambda i + 3j - 6k$ and the vector $i$ is $120^\circ$ . Find the exact value of the consta…
29Relative to the origin $O$ , two fixed points, $A$ and $B$ , have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectiv…
30A player is controlling an avatar in a virtual reality game where points $(x,y,z)$ are defined relative to a fixed point…
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 points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. $C$ is the point on line $OB$ s…
33Vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ are such that $\mathbf{b} \ne 0$ and $3\mathbf{a} \times \mathbf{b} = 7\mathbf{b} \times \mathbf{c}$ …
34Referred to the origin $O$ , the points $A$ , $B$ and $C$ have position vectors $3\mathbf{i} - \mathbf{j}$ , $5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ …
35The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).\n\nThe points A and B on$ …
36The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a} = \mathbf{i} + (\sin\theta)\mathbf{j} + (\cos\theta)\mathbf{k} \quad \text{and} \quad \mathbf{b} = \mathbf{i} + (\sin\phi)\mathbf{j} + (\cos\phi)\mathbf{k} \text{, where } 0 \le \theta \le \phi \le \pi .$ …
37Relative to the origin $O$ , three distinct fixed points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ …
38The molecular geometry is the three-dimensional arrangement of atoms in a molecule. Fig. 1 shows the molecular geometry…
39Relative to an origin $O$ , the points $A$ and $B$ have position vectors $4\mathbf{i} + 2\mathbf{j}$ and $3\mathbf{i} + \mathbf{k}$ …
40With reference to the point $O$ as the origin and the $x$ - $y$ plane as a horizontal plane, the pyramid $OPQRV$ has a par…
41The diagram above shows a cuboid $OABCDEFG$ with horizontal base $OABC$ where $OA = 6\text{ cm}$ and $AB = 4\text{ cm}$ .…
42Relative to the origin $O$ , the point $A$ has position vector $3\mathbf{i} + 7\mathbf{j} + \mathbf{k}$ . The plane $\pi$ …
43For positive real numbers $a$ , and $b$ , the lines $l_1$ and $l_2$ have equations given by $l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 0 \\ 3 \end{pmatrix} \text{ for } \lambda \in \mathbb{R} \quad \text{and} \quad l_2: \mathbf{r} = \begin{pmatrix} a \\ 9 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 7 \\ b \end{pmatrix} \text{ for } \mu \in \mathbb{R}.$ …
44Relative to the origin $O$ , two points $A$ and $B$ have position vectors given by $\mathbf{a} = 4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ …
45The 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…
46With reference to the point $O$ as the origin and the x-y plane as a horizontal plane, the pyramid $OPQRV$ has a paralle…
47Referred to the origin $O$ , the position vectors of three points $A$ , $B$ and $P$ are $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{a} + 5\mathbf{b}$ …
48The position vectors of the points $A$ , $B$ and $C$ with respect to the origin $O$ are $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ …
49The equations of four planes $p_{1}$ , $p_{2}$ , $p_{3}$ and $p_{4}$ are $x+y+z=2,\\-x+ay+z=1,\\2x+2y+2z=b,\\cx+2y+3z=d,$ …