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