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 \(\vec{AB}\).
Відповідь
(C)
\(\begin{pmatrix} -1 \\ -2 \end{pmatrix}\)
2
Which of the following is the correct method for finding the magnitude of the vector \(\begin{pmatrix} x \\ y \end{pmatrix}\)?
Відповідь
(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 \(2\vec{a} - \vec{b}\)?
Відповідь
(B)
\(\begin{pmatrix} 5 \\ 5 \end{pmatrix}\)
4
What does the top number in a column vector indicate?
Відповідь
(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 \(\vec{q}\).
Відповідь
(B)
\(\vec{p} + \vec{q}\)
6
If \(\vec{p} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\), what is - \(\frac{1}{2}\vec{p}\)?
Відповідь
(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 \(2\vec{u} + \vec{w}\).
Відповідь
(C)
\(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\)
9
The notation \[\begin{bmatrix} x \\ y \end{bmatrix}\] is often used to represent:
Відповідь
(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\(\vec{a}\) + \(\vec{b}\)?
Відповідь
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?
Відповідь
(B)
\(\begin{bmatrix} -4 \\ 6 \end{bmatrix}\)
12
If \(\vec{a} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\), what is the magnitude of 2\(\vec{a}\)?
Відповідь
(B)
\(2\sqrt{2}\)
13
In a triangle, \(\vec{AB} = \vec{p}\) and \(\vec{BC} = \vec{q}\). What is the value of \(\vec{AC}\)?
Відповідь
(B)
\(\vec{p} + \vec{q}\)
14
Which of the following operations is NOT defined for vectors?
Відповідь
(D)
Vector division
15
What is the magnitude of the vector \(\begin{pmatrix} 5 \\ -12 \end{pmatrix}\)?
Відповідь
(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 \( 3\vec{u} + 2\vec{v} \)?
Відповідь
(C)
\(\begin{bmatrix} 1 \\ 10 \end{bmatrix}\)
17
If \(\vec{u} = 4\vec{v}\), which statement is ALWAYS true?
Відповідь
(B)
\(\vec{u}\) and \(\vec{v}\) have the same direction.
18
If \(\vec{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\), what is -3\(\vec{a}\)?
Відповідь
(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?
Відповідь
(C)
\(\begin{bmatrix} 5 \\ 7 \end{bmatrix}\)
20
Given the points A(1, -1) and B(3, 2), the vector \(\vec{AB}\) is:
Відповідь
(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 \(\vec{AC}\)?
Відповідь
(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:
Відповідь
(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 \(\vec{a} + \vec{b}\)?
Відповідь
(B)
\(\sqrt{10}\)
24
If \( \vec{a} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \), what is \( -2\vec{a} \) equal to?
Відповідь
(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 \(\vec{BC}\)?
Відповідь
(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?
Відповідь
(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 \(\vec{a} + 2\vec{b}\)?
Відповідь
(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?
Відповідь
(A)
\(\vec{0}\)
29
If \(\vec{p} + \vec{q} = \vec{0}\), what is the relationship between \(\vec{p}\) and \(\vec{q}\)?
Відповідь
(B)
They are equal in magnitude and opposite in direction.
30
In the context of column vectors, what does the term 'scalar' represent?
Відповідь
(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?
Відповідь
(A)
[4, -2]
32
If two vectors are parallel, what does that tell you about their directions?
Відповідь
(C)
They have the same direction or opposite directions
33
What does the top number in a column vector represent?
Відповідь
(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?
Відповідь
(D)
All of the above are possible
35
If a shape is translated by the vector [4, -1], the shape moves:
Відповідь
(B)
4 units right and 1 unit down
36
If two vectors are parallel, what can be said about the relationship of their components?
Відповідь
(B)
Their components are proportional.
37
What type of quantity is the magnitude of a vector?
Відповідь
(B)
A scalar
38
What is the result of adding a vector to its negative?
Відповідь
(C)
The zero vector.
39
Which of the following is used for vector addition?
Відповідь
(A)
Adding the x-components and y-components separately
40
If AB = [4, -2] and AC = [1, 3], what is the vector CB?
Відповідь
(B)
[-3, 5]
41
What is the term used to describe a vector whose direction can be reversed?
Відповідь
(B)
Opposite vector
42
If a = [1, 2] and b = [-3, 1], what is the magnitude of the vector 2a + b?
Відповідь
(C)
5
43
If vector AB is [3, 2], and we want to find the vector BA, what is the new vector?
Відповідь
(C)
[-3, -2]
44
What can you conclude about the vectors AB and CD if AB = CD?
Відповідь
(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?
Відповідь
(A)
[4, 1]
46
If the position vectors of points P and Q are p and q respectively, what represents the vector QP?
Відповідь
(C)
p - q
47
Which of the following statements is FALSE regarding vector notation?
Відповідь
(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?
Відповідь
(B)
a + b
49
If a vector 'v' is [1, 0], and is multiplied by 0, what is the resulting vector?
