Mathematics for IGCSE & O level - Vectors (Section 2)
1
If \(\vec{OA} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\vec{OB} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\), find \(\vec{AB}\).
คำตอบ
(C)
\(\begin{pmatrix} -1 \\ -2 \end{pmatrix}\)
2
Which of the following is the correct method for finding the magnitude of the vector \(\begin{pmatrix} x \\ y \end{pmatrix}\)?
คำตอบ
(D)
\(\sqrt{x^2 + y^2}\)
3
If \(\vec{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}\), what is \(2\vec{a} - \vec{b}\)?
คำตอบ
(B)
\(\begin{pmatrix} 5 \\ 5 \end{pmatrix}\)
4
What does the top number in a column vector indicate?
คำตอบ
(B)
Movement left or right.
5
In triangle ABC, given that \(\vec{AB} = \vec{p}\) and \(\vec{BC} = \vec{q}\), express \(\vec{AC}\) in terms of \(\vec{p}\) and \(\vec{q}\).
คำตอบ
(B)
\(\vec{p} + \vec{q}\)
6
If \(\vec{p} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\), what is - \(\frac{1}{2}\vec{p}\)?
คำตอบ
(A)
\(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\)
7
If \(\vec{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\), which of the following expressions correctly represents the calculation for 3\(\vec{a}\) + 2\(\vec{b}\)?
Given \(\vec{u} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\) and \(\vec{w} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\), calculate \(2\vec{u} + \vec{w}\).
คำตอบ
(C)
\(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\)
9
The notation \[\begin{bmatrix} x \\ y \end{bmatrix}\] is often used to represent:
คำตอบ
(C)
A column vector.
10
Given \( ec{a} = egin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \( ec{b} = egin{pmatrix} 3 \\ -1 \end{pmatrix}\), which of the following is/are true about the vector 2\(\vec{a}\) + \(\vec{b}\)?
คำตอบ
A
C
11
If \( \vec{a} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} \) and \( \vec{b} = \begin{bmatrix} 3 \\ -4 \end{bmatrix} \), what is \( \vec{a} - \vec{b} \) equal to?
คำตอบ
(B)
\(\begin{bmatrix} -4 \\ 6 \end{bmatrix}\)
12
If \(\vec{a} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\), what is the magnitude of 2\(\vec{a}\)?
คำตอบ
(B)
\(2\sqrt{2}\)
13
In a triangle, \(\vec{AB} = \vec{p}\) and \(\vec{BC} = \vec{q}\). What is the value of \(\vec{AC}\)?
คำตอบ
(B)
\(\vec{p} + \vec{q}\)
14
Which of the following operations is NOT defined for vectors?
คำตอบ
(D)
Vector division
15
What is the magnitude of the vector \(\begin{pmatrix} 5 \\ -12 \end{pmatrix}\)?
คำตอบ
(B)
13
16
Given the vectors \( \vec{u} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( \vec{v} = \begin{bmatrix} -1 \\ 3 \end{bmatrix} \), which of the following represents \( 3\vec{u} + 2\vec{v} \)?
คำตอบ
(C)
\(\begin{bmatrix} 1 \\ 10 \end{bmatrix}\)
17
If \(\vec{u} = 4\vec{v}\), which statement is ALWAYS true?
คำตอบ
(B)
\(\vec{u}\) and \(\vec{v}\) have the same direction.
18
If \(\vec{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\), what is -3\(\vec{a}\)?
คำตอบ
(A)
\(\begin{pmatrix} -9 \\ 3 \end{pmatrix}\)
19
If \(\vec{a} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}\) and \(\vec{b} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\), then what is \( 2\vec{a} - \vec{b} \) equal to?
คำตอบ
(C)
\(\begin{bmatrix} 5 \\ 7 \end{bmatrix}\)
20
Given the points A(1, -1) and B(3, 2), the vector \(\vec{AB}\) is:
คำตอบ
(B)
\(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)
21
In a parallelogram, if \(\vec{AB} = \vec{a}\) and \(\vec{AD} = \vec{b}\), which of the following is equal to \(\vec{AC}\)?
คำตอบ
(B)
\(\vec{a} + \vec{b}\)
22
Given \(\vec{AB} = 2\vec{i} + 3\vec{j}\) and \(\vec{BC} = -\vec{i} + \vec{j}\), then \(\vec{AC}\) is:
คำตอบ
(A)
\(\vec{i} + 4\vec{j}\)
23
If \(\vec{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), and \(\vec{b} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\), what is the magnitude of \(\vec{a} + \vec{b}\)?
คำตอบ
(B)
\(\sqrt{10}\)
24
If \( \vec{a} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \), what is \( -2\vec{a} \) equal to?
คำตอบ
(A)
\(\begin{bmatrix} -6 \\ 4 \end{bmatrix}\)
25
In the triangle ABC, if \(\vec{AB} = \vec{p}\) and \(\vec{AC} = \vec{q}\), which of the following represents \(\vec{BC}\)?
คำตอบ
(B)
\(\vec{q} - \vec{p}\)
26
If a shape is translated by the vector \[\begin{bmatrix} 2 \\ -3 \end{bmatrix}\], how does the shape move?
คำตอบ
(C)
2 units to the right and 3 units down
27
Given \(\vec{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\), which of the following represents \(\vec{a} + 2\vec{b}\)?
คำตอบ
(A)
\(\begin{pmatrix} 4 \\ 7 \end{pmatrix}\)
28
In triangle ABC, if \(\vec{AB} = \vec{u}\), \(\vec{BC} = \vec{v}\), and \(\vec{CA} = \vec{w}\), then what is \(\vec{u} + \vec{v} + \vec{w}\) equal to?
คำตอบ
(A)
\(\vec{0}\)
29
If \(\vec{p} + \vec{q} = \vec{0}\), what is the relationship between \(\vec{p}\) and \(\vec{q}\)?
คำตอบ
(B)
They are equal in magnitude and opposite in direction.
30
In the context of column vectors, what does the term 'scalar' represent?
คำตอบ
(B)
A number that multiplies a vector.
31
If a vector 'a' represents the translation of a point to the right 4 units and down 2 units, what is the vector?
คำตอบ
(A)
[4, -2]
32
If two vectors are parallel, what does that tell you about their directions?
คำตอบ
(C)
They have the same direction or opposite directions
33
What does the top number in a column vector represent?
คำตอบ
(A)
Movement in the x-direction
34
Given that the magnitude of vector a is 4 and the magnitude of vector b is 3, what is a possible magnitude of a + b?
คำตอบ
(D)
All of the above are possible
35
If a shape is translated by the vector [4, -1], the shape moves:
คำตอบ
(B)
4 units right and 1 unit down
36
If two vectors are parallel, what can be said about the relationship of their components?
คำตอบ
(B)
Their components are proportional.
37
What type of quantity is the magnitude of a vector?
คำตอบ
(B)
A scalar
38
What is the result of adding a vector to its negative?
คำตอบ
(C)
The zero vector.
39
Which of the following is used for vector addition?
คำตอบ
(A)
Adding the x-components and y-components separately
40
If AB = [4, -2] and AC = [1, 3], what is the vector CB?
คำตอบ
(B)
[-3, 5]
41
What is the term used to describe a vector whose direction can be reversed?
คำตอบ
(B)
Opposite vector
42
If a = [1, 2] and b = [-3, 1], what is the magnitude of the vector 2a + b?
คำตอบ
(C)
5
43
If vector AB is [3, 2], and we want to find the vector BA, what is the new vector?
คำตอบ
(C)
[-3, -2]
44
What can you conclude about the vectors AB and CD if AB = CD?
คำตอบ
(B)
AB and CD are equal in magnitude and direction.
45
If the vector AB is [3, -1] and the vector BC is [1, 2], what is the vector AC?
คำตอบ
(A)
[4, 1]
46
If the position vectors of points P and Q are p and q respectively, what represents the vector QP?
คำตอบ
(C)
p - q
47
Which of the following statements is FALSE regarding vector notation?
คำตอบ
(B)
Vectors always start at the origin
48
In a parallelogram ABCD, with vectors AB = a and AD = b, which vector is equivalent to AC?
คำตอบ
(B)
a + b
49
If a vector 'v' is [1, 0], and is multiplied by 0, what is the resulting vector?
