Math topic中文

Addition subtraction scalar multiplication

Addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations.

Practice questions

1A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
2The equations of planes $p_1$ and $p_2$ are given by $p_1: \mathbf{r} \cdot \begin{pmatrix} \alpha \\ 7 \\ 2 \end{pmatrix} = \alpha + 20, \; \alpha \in \mathbb{R},$ …
3In the triangle shown below, one vertex is origin O, and the two other vertices are A and B where $\overrightarrow{OA}=\mathbf{a}$ …
4Referred 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…
5Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ …
6The points A, B and C have position vectors $2i,\;3j$ and $4k$ respectively. Find a vector which is perpendicular to the…
7OABC is a rhombus. The points A and C are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$ …
8In triangle $PQR$ , the vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ represent the vectors $\vec{PQ}$ , $\vec{QR}$ …
9Relative to the origin $O$ , points $A$ , $B$ and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{c}$ re…
10It 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…
11Relative to the origin $O$ , the points $A$ , $B$ , and $C$ have position vectors $\mathbf{a}$ , $\mathbf{b}$ and $3\mathbf{a}+\mathbf{b}$ …
12The planes $p_1$ and $p_2$ have equations $\mathbf{r} = \begin{pmatrix} 4 \\ -1 \\ 3 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 7 \end{pmatrix} + t \begin{pmatrix} 1 \\ -3 \\ 12 \end{pmatrix}$ …
13With reference to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectivel…
14With respect to the origin $O$ , the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respect…
15To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
16With reference to the origin $O$ , the position vectors of points $A$ , $B$ and $C$ are expressed as $\overrightarrow{OA}$ …
17Methane ( CH_4 ) is a chemical compound with a tetrahedral structure. The 4 hydrogen (H) atoms form a regular tetrahedro…
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 $O$ , the position vectors of points $A$ and $B$ are $\mathbf{a}$ and $\mathbf{b}$ respectively, w…
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 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…
22Referred to an origin $O$ , the position vectors of three points $A$ , $B$ and $C$ are $3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$ …
23The position vector of point X is $\begin{pmatrix}8\\-4\end{pmatrix}$ . The position vector of point Y is $\begin{pmatrix}6\\4\end{pmatrix}$ …
246 equilateral triangles each with sides 5 units are arranged to form a regular hexagon as shown in the diagram below. By…
25With reference to the origin $O$ , the points $A$ , $B$ and $C$ are such that $\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ …
26With reference to the origin O, the point A has position vector $-\mathbf{i} - 3\mathbf{j} + 3\mathbf{k}$ . The plane $\pi$ …
27Referred to the origin O, let P, Q and R be distinct points with position vectors $\mathbf{p}$ , $\mathbf{q}$ and $\mathbf{r}$ …
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 $\Pi_2$ have equations $\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) = 4$ and $\mathbf{r} = 3\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + \mathbf{k}) + \mu(4\mathbf{i} + \mathbf{j})$ …
30A 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)…
31The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. $C$ is the point on line $OB$ s…
32The position vectors of $A$ and $B$ with respect to the origin $O$ are $\mathbf{a}$ and $\mathbf{b}$ respectively. The p…
33Referred to the origin $O$ , the points $A$ , $B$ and $C$ have position vectors $3\mathbf{i} - \mathbf{j}$ , $5\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}$ …
3424 In the grid, $\overrightarrow{OA} = a$ and $\overrightarrow{OB} = b$ . Q is a point on the grid. Mark and label the po…
35The plane $\Pi$ has equation $2x-3y+6z=7$ and the line $l$ has equation $\mathbf{r}=8\mathbf{i}-2\mathbf{j}+8\mathbf{k}+t(3\mathbf{i}-\mathbf{j}+2\mathbf{k}).\n\nThe points A and B on$ …
36Referred to the origin $O$ , points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively such tha…
37Relative to an origin $O$ , the points $A$ and $B$ have position vectors $4\mathbf{i} + 2\mathbf{j}$ and $3\mathbf{i} + \mathbf{k}$ …
38With reference to the point $O$ as the origin and the $x$ - $y$ plane as a horizontal plane, the pyramid $OPQRV$ has a par…
39Relative to the origin $O$ , the point $A$ has position vector $3\mathbf{i} + 7\mathbf{j} + \mathbf{k}$ . The plane $\pi$ …
40The diagram shows a unit circle with centre $O$ and diameter $AB$ . The point $C$ lies on the circumference of the circle…
41As shown in the above parallelogram $OACB$ , $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . I…
42The non-zero vectors $\mathbf{a}$ , $\mathbf{b}$ , and $\mathbf{c}$ are such that $3\mathbf{a}\times\mathbf{b} = 2\mathbf{b}\times\mathbf{c}$ …
43The position vectors of A and B with respect to the origin O are $\mathbf{a}$ and $\mathbf{b}$ respectively. The point C…
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…
46The non-zero vectors $\mathbf{a}$ , $\mathbf{b}$ , and $\mathbf{c}$ are such that $3\mathbf{a} \times \mathbf{b} = 2\mathbf{b} \times \mathbf{c}$ …
47The tangent at the point $(x, y)$ on the curve with equation $y = \mathrm{f}(x)$ passes through the point $(a, x)$ , wher…
48The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ with respect to origin $O$ , where $\mathbf{a}$ …
49Referred to the origin $O$ , the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ …
50Referred to the origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, whe…
51The diagram below shows a triangular base pyramid with vertices, $A$ , $B$ , $C$ and $D$ . With reference to the origin $O$ …
52With respect to an origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively w…
53With reference to the point $O$ as the origin and the x-y plane as a horizontal plane, the pyramid $OPQRV$ has a paralle…
54Referred to the origin $O$ , the position vectors of three points $A$ , $B$ and $P$ are $\mathbf{a}$ , $\mathbf{b}$ and $\mathbf{a} + 5\mathbf{b}$ …