Vectors

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Ghi chú bài học

Vectors and notation

A vector has both magnitude (size) and direction. Written as a column \begin{pmatrix} a \\ b \end{pmatrix}, or \vec{AB}, or in bold \mathbf{a}.
\vec{AB} goes from A to B. Its negative reverses direction: \vec{BA} = -\vec{AB}.
The zero vector has magnitude 0 and no defined direction. Statements about parallel directions here assume nonzero vectors.

Vector magnitude from components

Operations with vectors

Add vectors by adding components: \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}.
Multiply by a scalar to scale: 3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}.
The magnitude of \begin{pmatrix} a \\ b \end{pmatrix} is \√{a² + b²}. For \begin{pmatrix} 3 \\ 4 \end{pmatrix} it is 5.

Parallel vectors

Two nonzero vectors are parallel if one is a nonzero scalar multiple of the other. Positive multiples point the same way; negative multiples point in opposite directions.

Geometry with vectors

Use the triangle rule: \vec{AC} = \vec{AB} + \vec{BC}. Route to a point by any path along known vectors.
The midpoint M of AB satisfies \vec{AM} = \frac{1}{2}\vec{AB}.
Vectors prove that points are collinear (on a straight line): if \vec{AB} is a multiple of \vec{AC} then A, B and C are collinear.

Adding vectors head to tail

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 32 câu hỏi. Đăng ký để xem tất cả.

1.What two properties does a vector have?
Easy
- Asize only
- Bmagnitude and direction
- Cdirection only
- Dspeed and time
2.What is 2\begin{pmatrix} 3 \\ -1 \end{pmatrix}?
Easy
- A\begin{pmatrix} 6 \\ -2 \end{pmatrix}
- B\begin{pmatrix} 5 \\ 1 \end{pmatrix}
- C\begin{pmatrix} 6 \\ -1 \end{pmatrix}
- D\begin{pmatrix} 3 \\ -2 \end{pmatrix}
3.What is \begin{pmatrix} 4 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ 5 \end{pmatrix}?
Easy
- A\begin{pmatrix} 6 \\ 6 \end{pmatrix}
- B\begin{pmatrix} 8 \\ 5 \end{pmatrix}
- C\begin{pmatrix} 2 \\ -4 \end{pmatrix}
- D\begin{pmatrix} 6 \\ 4 \end{pmatrix}
4.What is \begin{pmatrix} 5 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix}?
Easy
- A\begin{pmatrix} 4 \\ -2 \end{pmatrix}
- B\begin{pmatrix} 6 \\ 6 \end{pmatrix}
- C\begin{pmatrix} 4 \\ 2 \end{pmatrix}
- D\begin{pmatrix} -4 \\ 2 \end{pmatrix}
5.What is -\begin{pmatrix} 2 \\ -3 \end{pmatrix}?
Easy
- A\begin{pmatrix} -2 \\ 3 \end{pmatrix}
- B\begin{pmatrix} 2 \\ 3 \end{pmatrix}
- C\begin{pmatrix} -2 \\ -3 \end{pmatrix}
- D\begin{pmatrix} 3 \\ -2 \end{pmatrix}
6.If \vec{AB} = \mathbf{a}, what is \vec{BA}?
Easy
- A\mathbf{a}
- B-\mathbf{a}
- C2\mathbf{a}
- D0
7.A has coordinates (1, 2) and B (4, 6). What is \vec{AB}?
Easy
- A\begin{pmatrix} 3 \\ 4 \end{pmatrix}
- B\begin{pmatrix} 5 \\ 8 \end{pmatrix}
- C\begin{pmatrix} -3 \\ -4 \end{pmatrix}
- D\begin{pmatrix} 4 \\ 3 \end{pmatrix}
8.What is the magnitude of \begin{pmatrix} 6 \\ 8 \end{pmatrix}?
Easy
- A10
- B14
- C48
- D100

Thêm câu hỏi

9.Which vector is parallel to \begin{pmatrix} 2 \\ 3 \end{pmatrix}?
Medium
- A\begin{pmatrix} 4 \\ 6 \end{pmatrix}
- B\begin{pmatrix} 3 \\ 2 \end{pmatrix}
- C\begin{pmatrix} 2 \\ 4 \end{pmatrix}
- D\begin{pmatrix} 6 \\ 8 \end{pmatrix}
10.If \mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}, what is 2\mathbf{a} + \mathbf{b}?
Medium
- A\begin{pmatrix} 5 \\ 3 \end{pmatrix}
- B\begin{pmatrix} 4 \\ 1 \end{pmatrix}
- C\begin{pmatrix} 5 \\ 1 \end{pmatrix}
- D\begin{pmatrix} 7 \\ 3 \end{pmatrix}
11.What is the magnitude of \begin{pmatrix} 5 \\ 12 \end{pmatrix}?
Medium
- A13
- B17
- C60
- D169
12.In triangle ABC, \vec{AB} = \mathbf{a} and \vec{BC} = \mathbf{b}. What is \vec{AC}?
Medium
- A\mathbf{a} + \mathbf{b}
- B\mathbf{a} - \mathbf{b}
- C\mathbf{b} - \mathbf{a}
- D\mathbf{a}\mathbf{b}
13.M is the midpoint of AB and \vec{AB} = \mathbf{a}. What is \vec{AM}?
Medium
- A\frac{1}{2}\mathbf{a}
- B2\mathbf{a}
- C-\mathbf{a}
- D\mathbf{a}
14.What is \vec{AC} if \vec{AB} = \mathbf{a} and \vec{CB} = \mathbf{b}?
Medium
- A\mathbf{a} - \mathbf{b}
- B\mathbf{a} + \mathbf{b}
- C\mathbf{b} - \mathbf{a}
- D-\mathbf{a} - \mathbf{b}
15.\vec{OA} = \mathbf{a}, \vec{OB} = \mathbf{b}. What is \vec{AB}?
Medium
- A\mathbf{b} - \mathbf{a}
- B\mathbf{a} - \mathbf{b}
- C\mathbf{a} + \mathbf{b}
- D-\mathbf{a} - \mathbf{b}
16.\vec{OA} = \mathbf{a}, \vec{OB} = \mathbf{b}, and M is the midpoint of AB. What is \vec{OM}?
Medium
- A\frac{1}{2}(\mathbf{a} + \mathbf{b})
- B\frac{1}{2}(\mathbf{b} - \mathbf{a})
- C\mathbf{a} + \mathbf{b}
- D\frac{1}{2}\mathbf{a}
17.\vec{AB} = \begin{pmatrix} 2 \\ 4 \end{pmatrix} and \vec{AC} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}. What can you say about A, B and C?
Medium
- Acollinear
- Bthey form a right angle
- CAB is longer
- Dno relation
18.For which k are \begin{pmatrix} k \\ 6 \end{pmatrix} and \begin{pmatrix} 2 \\ 3 \end{pmatrix} parallel?
Medium
- A4
- B6
- C3
- D9
19.What does \begin{pmatrix} 3 \\ -2 \end{pmatrix} represent as a movement?
Easy
- A3 right, 2 down
- B3 left, 2 up
- C3 up, 2 right
- D2 left, 3 down
20.If \mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, what is 3\mathbf{a}?
Easy
- A\begin{pmatrix} 6 \\ 3 \end{pmatrix}
- B\begin{pmatrix} 5 \\ 4 \end{pmatrix}
- C\begin{pmatrix} 6 \\ 1 \end{pmatrix}
- D\begin{pmatrix} 2 \\ 3 \end{pmatrix}
21.Two nonzero vectors are parallel when:
Medium
- Aone is a scalar multiple of the other
- Bthey have equal sums
- Cthey are equal in magnitude
- Dthey are at right angles
22.The route from A to C via B shows:
Medium
- A\vec{AC} = \vec{AB} + \vec{BC}
- B\vec{AC} = \vec{AB} - \vec{BC}
- C\vec{AC} = \vec{BC}
- D\vec{AC} = 0
23.A vector has both magnitude and direction.
Easy
True or false?
24.\vec{BA} = -\vec{AB}.
Easy
True or false?
25.The magnitude of \begin{pmatrix} 3 \\ 4 \end{pmatrix} is 7.
Easy
True or false?
26.Nonzero parallel vectors are scalar multiples of each other.
Easy
True or false?
27.For distinct A, B and C, if AB and AC are parallel vectors, then A, B and C are collinear.
Medium
True or false?
28.Which are parallel to \begin{pmatrix} 1 \\ 2 \end{pmatrix}?
Medium
- A\begin{pmatrix} 2 \\ 4 \end{pmatrix}
- B\begin{pmatrix} -3 \\ -6 \end{pmatrix}
- C\begin{pmatrix} 2 \\ 1 \end{pmatrix}
- D\begin{pmatrix} 3 \\ 6 \end{pmatrix}
- E\begin{pmatrix} 1 \\ 3 \end{pmatrix}
29.Which are true?
Medium
- A\vec{AC} = \vec{AB} + \vec{BC}
- B\vec{AB} = -\vec{BA}
- Cmagnitude \begin{pmatrix} 3 \\ 4 \end{pmatrix} = 5
- D2\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}
- Evectors have no direction
30.What is \begin{pmatrix} 1 \\ 5 \end{pmatrix} + \begin{pmatrix} 3 \\ 2 \end{pmatrix}?
Easy
- A\begin{pmatrix} 4 \\ 7 \end{pmatrix}
- B\begin{pmatrix} 3 \\ 10 \end{pmatrix}
- C\begin{pmatrix} 4 \\ 3 \end{pmatrix}
- D\begin{pmatrix} 2 \\ 3 \end{pmatrix}
31.What is 4 × \begin{pmatrix} 0 \\ 2 \end{pmatrix}?
Easy
- A\begin{pmatrix} 0 \\ 8 \end{pmatrix}
- B\begin{pmatrix} 4 \\ 6 \end{pmatrix}
- C\begin{pmatrix} 0 \\ 6 \end{pmatrix}
- D\begin{pmatrix} 4 \\ 8 \end{pmatrix}
32.\vec{OA} = \mathbf{a} and \vec{OB} = \mathbf{b}. M is the midpoint of AB. Find \vec{OM} in terms of \mathbf{a} and \mathbf{b}.
Medium

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