Vectors

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課程筆記

Introduction to Column Vectors

A column vector describes a translation: e.g. \begin{pmatrix}6\\3\end{pmatrix} means 6 right, 3 up.
Add/subtract vectors component-wise: top numbers together, bottom numbers together.
Multiply a vector by a scalar (a number) by multiplying each component.
Follow order of operations when combining scalar multiplication and addition.
Example: 2\begin{pmatrix}5\\2\end{pmatrix}+5\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}10\\4\end{pmatrix}+\begin{pmatrix}15\\-5\end{pmatrix}=\begin{pmatrix}25\\-1\end{pmatrix}.

Column vector components

Representing Vectors as Diagrams

A vector has magnitude (size) and direction; shown by an arrow.
Vectors are written in bold (or underlined by hand), e.g. \mathbf{a} or \underline{a}.
\overrightarrow{AB} goes from A to B; \overrightarrow{BA} is opposite direction.
To draw \begin{pmatrix}3\\4\end{pmatrix}, start at a point, move 3 right, 4 up, draw arrow.
Multiplying by a positive scalar changes length but not direction; negative scalar reverses direction.
To add vectors diagrammatically, place them tip-to-tail; the resultant goes from start to end.

Drawing vectors a, b and c on a grid

Magnitude of a Vector

The magnitude (modulus) of a vector is its length, always positive.
For \mathbf{a}=\begin{pmatrix}x\\y\end{pmatrix}, |\mathbf{a}|=\√{x²+y²} (Pythagoras).
Magnitude is independent of direction: |\overrightarrow{AB}|=|\overrightarrow{BA}|.
If a vector is multiplied by scalar k, its magnitude is multiplied by |k|.

The magnitude of a vector as the hypotenuse PQ

Position & Displacement Vectors

A position vector locates a point relative to the origin O: \overrightarrow{OA}=\mathbf{a}.
Coordinates equal components: point (3,-2) has position vector \begin{pmatrix}3\\-2\end{pmatrix}.
A displacement vector goes from one point to another: \overrightarrow{AB}=\mathbf{b}-\mathbf{a}.
Use \overrightarrow{AB}=-\mathbf{a}+\mathbf{b} to find displacement from position vectors.

Position and displacement vectors in triangle OAB

Finding Vector Paths

A vector path is a sequence of vectors from start to end.
In a grid of parallelograms, horizontal moves are multiples of one vector, diagonal moves of another.
Count steps in each direction to express a vector in terms of given vectors.
Negative signs indicate opposite direction.

A grid of parallelograms for vector paths

Problem Solving with Vectors

Two vectors are parallel if one is a scalar multiple of the other.
To prove three points are collinear, show two vectors between them are parallel and share a common point.
Ratios along a line: if AX:XB=3:5, then \overrightarrow{AX}=\frac{3}{8}\overrightarrow{AB}.
Use vector algebra to find unknown points or prove geometric properties.

Trapezium OABC

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練習題

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1.What is the column vector representing a translation of 3 units to the right and 4 units down?
Easy
- A\begin{pmatrix}3\\-4\end{pmatrix}
- B\begin{pmatrix}-3\\4\end{pmatrix}
- C\begin{pmatrix}3\\4\end{pmatrix}
- D\begin{pmatrix}-3\\-4\end{pmatrix}
2.Given \mathbf{a} = \begin{pmatrix}5\\2\end{pmatrix} and \mathbf{b} = \begin{pmatrix}3\\-1\end{pmatrix}, find \mathbf{a} + \mathbf{b}.
Easy
- A\begin{pmatrix}8\\1\end{pmatrix}
- B\begin{pmatrix}8\\3\end{pmatrix}
- C\begin{pmatrix}2\\1\end{pmatrix}
- D\begin{pmatrix}2\\3\end{pmatrix}
3.Given \mathbf{a} = \begin{pmatrix}4\\-2\end{pmatrix}, what is 2\mathbf{a}?
Easy
- A\begin{pmatrix}8\\-4\end{pmatrix}
- B\begin{pmatrix}6\\0\end{pmatrix}
- C\begin{pmatrix}8\\4\end{pmatrix}
- D\begin{pmatrix}2\\-1\end{pmatrix}
4.Find the magnitude of the vector \begin{pmatrix}3\\4\end{pmatrix}.
Easy
- A5
- B7
- C12
- D25
5.The displacement vector from A to B is \overrightarrow{AB} = \mathbf{b} - \mathbf{a}. If \mathbf{a} = \begin{pmatrix}2\\5\end{pmatrix} and \mathbf{b} = \begin{pmatrix}7\\1\end{pmatrix}, what is \overrightarrow{AB}?
Easy
- A\begin{pmatrix}5\\-4\end{pmatrix}
- B\begin{pmatrix}5\\4\end{pmatrix}
- C\begin{pmatrix}-5\\4\end{pmatrix}
- D\begin{pmatrix}9\\6\end{pmatrix}
6.Given \mathbf{p} = \begin{pmatrix}4\\5\end{pmatrix} and \mathbf{q} = \begin{pmatrix}-2\\7\end{pmatrix}, find 2\mathbf{p} + \mathbf{q}.
Easy
- A\begin{pmatrix}6\\17\end{pmatrix}
- B\begin{pmatrix}6\\-3\end{pmatrix}
- C\begin{pmatrix}2\\12\end{pmatrix}
- D\begin{pmatrix}10\\12\end{pmatrix}
7.Given \mathbf{a} = \begin{pmatrix}-3\\2\end{pmatrix}, find |\mathbf{a}|.
Easy
- A\√{13}
- B\√{5}
- C13
- D5
8.Point A is (6,4) and point B is (2,7). Write \overrightarrow{AB} as a column vector.
Easy
- A\begin{pmatrix}-4\\3\end{pmatrix}
- B\begin{pmatrix}4\\-3\end{pmatrix}
- C\begin{pmatrix}-4\\-3\end{pmatrix}
- D\begin{pmatrix}4\\3\end{pmatrix}

更多題目

9.Given \mathbf{u} = \begin{pmatrix}3\\-2\end{pmatrix} and \mathbf{v} = \begin{pmatrix}-12\\5\end{pmatrix}, find \mathbf{u} - 2\mathbf{v}.
Medium
- A\begin{pmatrix}27\\-12\end{pmatrix}
- B\begin{pmatrix}-21\\8\end{pmatrix}
- C\begin{pmatrix}27\\8\end{pmatrix}
- D\begin{pmatrix}-21\\-12\end{pmatrix}
10.Find the magnitude of \mathbf{v} = \begin{pmatrix}-12\\5\end{pmatrix}.
Easy
- A13
- B17
- C7
- D169
11.In triangle OAB, \overrightarrow{OA} = \mathbf{a} and \overrightarrow{OB} = \mathbf{b}. M is the midpoint of AB. Find \overrightarrow{OM} in terms of \mathbf{a} and \mathbf{b}.
Medium
- A\frac{1}{2}(\mathbf{a}+\mathbf{b})
- B\frac{1}{2}(\mathbf{b}-\mathbf{a})
- C\mathbf{a}+\frac{1}{2}\mathbf{b}
- D\frac{1}{2}\mathbf{a}+\mathbf{b}
12.Given \overrightarrow{GH} = \frac{5}{6}(2\mathbf{p}+\mathbf{q}) and \overrightarrow{JK} = \frac{5}{18}(2\mathbf{p}+\mathbf{q}), what is the geometrical relationship between \overrightarrow{GH} and \overrightarrow{JK}?
Medium
- AThey are parallel and in the same direction, with \overrightarrow{GH} three times as long as \overrightarrow{JK}.
- BThey are parallel and in opposite directions, with \overrightarrow{GH} three times as long as \overrightarrow{JK}.
- CThey are parallel and in the same direction, with \overrightarrow{JK} three times as long as \overrightarrow{GH}.
- DThey are not parallel.
13.O is the origin, \overrightarrow{OC} = \mathbf{c} and \overrightarrow{OD} = \mathbf{d}. T is a point on CD such that CT:TD = 2:1. M is the midpoint of TC. Find the position vector of M in terms of \mathbf{c} and \mathbf{d}.
Hard
- A\frac{1}{3}\mathbf{c} + \frac{2}{3}\mathbf{d}
- B\frac{2}{3}\mathbf{c} + \frac{1}{3}\mathbf{d}
- C\frac{1}{2}\mathbf{c} + \frac{1}{2}\mathbf{d}
- D\frac{1}{4}\mathbf{c} + \frac{3}{4}\mathbf{d}
14.P is (16,9) and Q is (22,24). N is on PQ such that PN:NQ = 2:1. Find the coordinates of N.
Medium
- A(20, 19)
- B(18, 14)
- C(20, 14)
- D(18, 19)
15.In triangle OAB, \overrightarrow{OA} = \mathbf{a}, \overrightarrow{OB} = \mathbf{b}. C is on OB such that OC:CB = 5:3. D is on AB such that AD:DB = 3:5. Find \overrightarrow{CD} in terms of \mathbf{a} and \mathbf{b}.
Hard
- A\frac{5}{8}\mathbf{a} - \frac{1}{4}\mathbf{b}
- B\frac{3}{8}\mathbf{a} + \frac{1}{4}\mathbf{b}
- C\frac{5}{8}\mathbf{a} + \frac{1}{4}\mathbf{b}
- D\frac{3}{8}\mathbf{a} - \frac{1}{4}\mathbf{b}
16.Given vector \mathbf{v} = \begin{pmatrix}10\\-24\end{pmatrix}, find |\mathbf{v}|.
Easy
- A26
- B34
- C√676
- D√(100−576)
17.Given \overrightarrow{AB} = \begin{pmatrix}6\\-1\end{pmatrix}, \overrightarrow{BC} = \begin{pmatrix}-2\\5\end{pmatrix}, and \overrightarrow{DC} = \begin{pmatrix}2\\-3\end{pmatrix}, find \overrightarrow{AC}.
Medium
- A\begin{pmatrix}4\\4\end{pmatrix}
- B\begin{pmatrix}8\\-6\end{pmatrix}
- C\begin{pmatrix}4\\-6\end{pmatrix}
- D\begin{pmatrix}8\\4\end{pmatrix}
18.Given \mathbf{a} = \begin{pmatrix}-3\\2\end{pmatrix} and \mathbf{b} = \begin{pmatrix}5\\4\end{pmatrix}, find 3\mathbf{a} - 2\mathbf{b}.
Medium
- A\begin{pmatrix}-19\\-2\end{pmatrix}
- B\begin{pmatrix}1\\14\end{pmatrix}
- C\begin{pmatrix}-19\\2\end{pmatrix}
- D\begin{pmatrix}1\\-2\end{pmatrix}
19.Given \overrightarrow{OA} = \begin{pmatrix}4\\3\end{pmatrix}, \overrightarrow{AB} = \begin{pmatrix}8\\-7\end{pmatrix}, and \overrightarrow{AC} = \begin{pmatrix}-3\\6\end{pmatrix}, find \overrightarrow{OB}.
Medium
- A\begin{pmatrix}12\\-4\end{pmatrix}
- B\begin{pmatrix}12\\-10\end{pmatrix}
- C\begin{pmatrix}4\\-4\end{pmatrix}
- D\begin{pmatrix}12\\4\end{pmatrix}
20.Find the magnitude of the vector \begin{pmatrix}-1\\7\end{pmatrix}.
Easy
- A5\√{2}
- B\√{50}
- C8
- D√(1+49)
21.Given \mathbf{m} = \begin{pmatrix}5\\7\end{pmatrix}, find 3\mathbf{m}.
Easy
- A\begin{pmatrix}15\\21\end{pmatrix}
- B\begin{pmatrix}8\\10\end{pmatrix}
- C\begin{pmatrix}15\\10\end{pmatrix}
- D\begin{pmatrix}8\\21\end{pmatrix}
22.Given \mathbf{u} = \begin{pmatrix}7\\-4\end{pmatrix} and \mathbf{v} = \begin{pmatrix}3\\-2\end{pmatrix}, find 2\mathbf{u} - \mathbf{v}.
Medium
- A\begin{pmatrix}11\\-6\end{pmatrix}
- B\begin{pmatrix}11\\-2\end{pmatrix}
- C\begin{pmatrix}17\\-6\end{pmatrix}
- D\begin{pmatrix}17\\-2\end{pmatrix}
23.What is a column vector?
Easy
- AA vector that describes a translation using a horizontal and vertical component
- BA vector that only has a magnitude
- CA vector that only has a direction
- DA vector that cannot be multiplied by a scalar
24.The vector \begin{pmatrix}4\\-3\end{pmatrix} means:
Easy
- A4 units to the right and 3 units down
- B4 units to the right and 3 units up
- C4 units to the left and 3 units down
- D4 units to the left and 3 units up
25.What is the result of \begin{pmatrix}2\\5\end{pmatrix} + \begin{pmatrix}3\\-1\end{pmatrix}?
Easy
- A\begin{pmatrix}5\\4\end{pmatrix}
- B\begin{pmatrix}5\\6\end{pmatrix}
- C\begin{pmatrix}5\\-4\end{pmatrix}
- D\begin{pmatrix} -1\\6\end{pmatrix}
26.Given \mathbf{a} = \begin{pmatrix}3\\-2\end{pmatrix}, what is -\mathbf{a}?
Easy
- A\begin{pmatrix}-3\\2\end{pmatrix}
- B\begin{pmatrix}-3\\-2\end{pmatrix}
- C\begin{pmatrix}3\\2\end{pmatrix}
- D\begin{pmatrix} -2\\3\end{pmatrix}
27.The vector \begin{pmatrix}0\\5\end{pmatrix} represents:
Easy
- A5 units vertically, no horizontal movement
- B5 units horizontally, no vertical movement
- CA vector of length 0
- DA vector pointing to the right
28.Find \begin{pmatrix}4\\-1\end{pmatrix} - \begin{pmatrix}2\\3\end{pmatrix}.
Easy
- A\begin{pmatrix}2\\-4\end{pmatrix}
- B\begin{pmatrix}2\\4\end{pmatrix}
- C\begin{pmatrix}6\\2\end{pmatrix}
- D\begin{pmatrix} -2\\-4\end{pmatrix}
29.Given \mathbf{a} = \begin{pmatrix}2\\-3\end{pmatrix} and \mathbf{b} = \begin{pmatrix}-1\\4\end{pmatrix}, find 2\mathbf{a} + \mathbf{b}.
Easy
- A\begin{pmatrix}3\\-2\end{pmatrix}
- B\begin{pmatrix}3\\2\end{pmatrix}
- C\begin{pmatrix}1\\-2\end{pmatrix}
- D\begin{pmatrix}4\\-6\end{pmatrix}
30.What is the magnitude of the vector \begin{pmatrix}-6\\8\end{pmatrix}?
Easy
- A10
- B14
- C100
- D2
31.If \mathbf{a} = \begin{pmatrix}3\\-4\end{pmatrix}, which vector is parallel to \mathbf{a}?
Easy
- A\begin{pmatrix}-6\\8\end{pmatrix}
- B\begin{pmatrix}6\\8\end{pmatrix}
- C\begin{pmatrix}-3\\-4\end{pmatrix}
- D\begin{pmatrix}9\\12\end{pmatrix}
32.Point A has coordinates (2,5) and point B has coordinates (7,3). Write \overrightarrow{AB} as a column vector.
Easy
- A\begin{pmatrix}5\\-2\end{pmatrix}
- B\begin{pmatrix}5\\2\end{pmatrix}
- C\begin{pmatrix}-5\\2\end{pmatrix}
- D\begin{pmatrix} -5\\-2\end{pmatrix}
33.If \mathbf{a} = \begin{pmatrix}5\\7\end{pmatrix}, find 3\mathbf{a}.
Easy
- A\begin{pmatrix}15\\21\end{pmatrix}
- B\begin{pmatrix}15\\10\end{pmatrix}
- C\begin{pmatrix}8\\10\end{pmatrix}
- D\begin{pmatrix}5\\21\end{pmatrix}
34.What does the column vector \begin{pmatrix}6\\3\end{pmatrix} represent?
Easy
- A6 units right and 3 units up
- B6 units left and 3 units down
- C3 units right and 6 units up
- D6 units right and 3 units down
35.Given \mathbf{a} = \begin{pmatrix}2\\-5\end{pmatrix}, what is -\mathbf{a}?
Easy
- A\begin{pmatrix}-2\\5\end{pmatrix}
- B\begin{pmatrix}2\\-5\end{pmatrix}
- C\begin{pmatrix}-2\\-5\end{pmatrix}
- D\begin{pmatrix}2\\5\end{pmatrix}
36.The magnitude of a vector is always non-negative.
Easy
True or false?
37.Given \mathbf{a} = \begin{pmatrix}4\\-3\end{pmatrix} and \mathbf{b} = \begin{pmatrix}2\\1\end{pmatrix}, find \mathbf{a} - \mathbf{b}.
Easy
- A\begin{pmatrix}2\\-4\end{pmatrix}
- B\begin{pmatrix}6\\-2\end{pmatrix}
- C\begin{pmatrix}2\\-2\end{pmatrix}
- D\begin{pmatrix}-2\\4\end{pmatrix}
38.Match each vector operation to its result.
Medium
- \begin{pmatrix}3\\4\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix}
- 2\begin{pmatrix}1\\-3\end{pmatrix}
- \begin{pmatrix}5\\0\end{pmatrix} - \begin{pmatrix}2\\3\end{pmatrix}
- \begin{pmatrix}4\\2\end{pmatrix}
- \begin{pmatrix}2\\-6\end{pmatrix}
- \begin{pmatrix}3\\-3\end{pmatrix}
39.Which of the following vectors are parallel to \mathbf{a} = \begin{pmatrix}2\\-3\end{pmatrix}? (Select all that apply)
Medium
- A\begin{pmatrix}4\\-6\end{pmatrix}
- B\begin{pmatrix}-2\\3\end{pmatrix}
- C\begin{pmatrix}6\\9\end{pmatrix}
- D\begin{pmatrix}2\\3\end{pmatrix}
40.Arrange the steps to find the magnitude of \mathbf{v} = \begin{pmatrix}3\\-4\end{pmatrix}.
Easy
- Square each component
- Add the squares
- Take the square root
- Write the vector
41.Match each vector operation with its correct result. All vectors are given as column vectors.
Medium
- \begin{pmatrix}2\\3\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix}
- 3\begin{pmatrix}1\\-4\end{pmatrix}
- \begin{pmatrix}5\\2\end{pmatrix} - \begin{pmatrix}2\\-1\end{pmatrix}
- \begin{pmatrix}3\\1\end{pmatrix}
- \begin{pmatrix}3\\-12\end{pmatrix}
- \begin{pmatrix}3\\3\end{pmatrix}
42.Arrange the following steps in the correct order to find the magnitude of vector \begin{pmatrix}3\\-4\end{pmatrix}.
Easy
- |\mathbf{v}| = \√{3² + (-4)²}
- |\mathbf{v}| = \√{9 + 16}
- |\mathbf{v}| = \√{25}
- |\mathbf{v}| = 5
43.Which of the following vectors are parallel to \mathbf{a} = \begin{pmatrix}2\\-3\end{pmatrix}? (Select all that apply.)
Medium
- A\begin{pmatrix}4\\-6\end{pmatrix}
- B\begin{pmatrix}-2\\3\end{pmatrix}
- C\begin{pmatrix}6\\-9\end{pmatrix}
- D\begin{pmatrix}4\\6\end{pmatrix}
- E\begin{pmatrix}1\\-1.5\end{pmatrix}
44.Find the magnitude of the vector \begin{pmatrix}5\\-12\end{pmatrix}.
Easy
45.Complete the sentence:
Easy
A vector has both ____ and ____.
46.Write down the column vector \overrightarrow{AB} if A = (1, 4) and B = (5, 2).
Easy
47.If \overrightarrow{OA} = \begin{pmatrix}3\\-2\end{pmatrix} and \overrightarrow{OB} = \begin{pmatrix}5\\1\end{pmatrix}, find |\overrightarrow{AB}|.
Medium
48.Find the magnitude of the vector \begin{pmatrix}6\\8\end{pmatrix}.
Easy
49.Write down the column vector \overrightarrow{AB} if A = (1, 4) and B = (5, 2).
Easy
50.Explain why \begin{pmatrix}2\\4\end{pmatrix} and \begin{pmatrix}1\\2\end{pmatrix} are parallel.
Medium
51.Find the magnitude of the vector \begin{pmatrix}5\\-12\end{pmatrix}.
Easy
- A15
- B12
- C14
- D13
52.Find the magnitude of the vector \begin{pmatrix}6\\8\end{pmatrix}.
Easy
- A11
- B9
- C12
- D10

複習筆記

## Vectors - IGCSE Maths (CIE) Extended ### Column Vectors - A column vector describes a translation: \(\begin{pmatrix} x \\ y \end{pmatrix}\) means \(x\) units right, \(y\) units up. - **Addition**: Add top and bottom numbers separately: \(\begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 8 \\ 1 \end{pmatrix}\). - **Subtraction**: Subtract components: \(\begin{pmatrix} 5 \\ 2 \end{pmatrix} - \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\). - **Scalar multiplication**: Multiply each component: \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\). ### Magnitude (Modulus) - Length of vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\): \(|\mathbf{a}| = \sqrt{x^2 + y^2}\). - Always positive; direction irrelevant. ### Position & Displacement Vectors - **Position vector** of point A from origin O: \(\overrightarrow{OA} = \mathbf{a}\). - **Displacement vector** from A to B: \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\). ### Vector Paths - Follow paths using given vectors (e.g., \(\mathbf{a}\), \(\mathbf{b}\)). - Opposite direction: \(-\mathbf{a}\). - Simplify by combining like terms. ### Problem Solving - **Parallel vectors**: One is scalar multiple of the other: \(\mathbf{b} = k\mathbf{a}\). - **Collinearity**: Show two vectors are parallel and share a common point. - **Ratios**: Convert to fractions. E.g., \(AP:PC = 3:1\) means \(\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AC}\). ### Key Formulas - \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\) - \(|\mathbf{a}| = \sqrt{x^2 + y^2}\) - Parallel: \(\mathbf{b} = k\mathbf{a}\)

Column Vector: A vector written as \(\begin{pmatrix} x \\ y \end{pmatrix}\) representing horizontal and vertical components.
Magnitude: The length of a vector, found using Pythagoras: \(\sqrt{x^2 + y^2}\).
Position Vector: The vector from the origin O to a point, e.g., \(\overrightarrow{OA} = \mathbf{a}\).
Displacement Vector: The vector from one point to another: \(\overrightarrow{AB} = \mathbf{b} - \mathbf{a}\).
Scalar: A number that multiplies a vector, changing its magnitude but not direction (unless negative).
Parallel Vectors: Vectors that are scalar multiples of each other: \(\mathbf{b} = k\mathbf{a}\).
Collinear: Points that lie on the same straight line; shown by parallel vectors sharing a point.

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