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