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