Mathematics for IGCSE & O level - Vectors (Section 1)
1
If \(\vec{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\), what is the magnitude of \(\vec{a}\)?
Answer
(C)
\(\sqrt{29}\)
2
In a triangle ABC, where D is the midpoint of BC, which of the following is/are true?
Answer
A
B
C
3
Consider the parallelogram OABC where \(OA = a\) and \(OC = c\). Which of the following vector equations are true?
Answer
A
C
4
If \(\vec{a} = 4\vec{b}\), what can you conclude about vectors \(\vec{a}\) and \(\vec{b}\)?
Answer
(B)
They have the same direction.
5
In a parallelogram OABC, \(\vec{OA} = \vec{a}\) and \(\vec{OC} = \vec{c}\). M is the midpoint of BC. Which of the following statements are true?
Answer
B
D
6
What does it mean to multiply a vector by a scalar?
Answer
(A)
To add the vector to itself a certain number of times
7
If \( \vec{a} = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \), and \( \vec{b} = 2\vec{a} \) , what are the components of \( \vec{b} \)?
Answer
(B)
\(\begin{bmatrix} 8 \\ 0 \end{bmatrix}\)
8
Which of the following statements is TRUE about the vector \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\) ?
Answer
(B)
It represents the zero vector.
9
A transformation that moves a point in a given direction for a given distance is called a...
Answer
(C)
Translation
10
If \(\vec{p} = \begin{pmatrix} 3 \\ 0 \end{pmatrix}\) and \(\vec{q} = \begin{pmatrix} 0 \\ 4 \end{pmatrix}\), then what is \(\vec{p} + \vec{q}\) ?
Answer
(A)
\(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
11
What would happen to a vector, when multiplied by a scalar of -1?
Answer
(C)
The magnitude would stay the same and the direction would change.
12
Which of the following is a correct property of vector addition?
Answer
A
B
D
13
Given the points A(1, 2) and B(4, 6), what is the vector \(\vec{AB}\) represented as a column vector?
Answer
(A)
\(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
14
Given the vectors \(\vec{u} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\), which of the following are true?
Answer
A
C
D
15
What is the result of \( \begin{bmatrix} 4 \\ 1 \end{bmatrix} + \begin{bmatrix} -2 \\ 3 \end{bmatrix} \)?
Answer
(B)
\(\begin{bmatrix} 2 \\ 4 \end{bmatrix}\)
16
Given two vectors \[\vec{p} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}\] and \[\vec{q} = \begin{bmatrix} 2 \\ -1 \end{bmatrix}\] what is the magnitude of \( \vec{p} + \vec{q} \)?
Answer
(C)
5
17
If \(\vec{AB} = 3\vec{a}\) and \(\vec{BC} = 2\vec{a}\), what is the value of \(\vec{AC}\)?
Answer
(B)
\(5\vec{a}\)
18
Given that a = \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\) and b = \(\begin{pmatrix} 1 \\ -1 \end{pmatrix}\), what is the result of 3a + b?
Answer
(A)
\(\begin{pmatrix} 4 \\ 2 \end{pmatrix}\)
19
Given that \(\vec{a} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\), then \(2\vec{a} - \vec{b}\) is equal to:
Answer
(B)
\(\begin{pmatrix} -5 \\ 5 \end{pmatrix}\)
20
If \(\vec{AB} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\), what is \(\vec{BA}\) ?
Answer
(B)
\(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\)
21
If \(3\vec{x} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}\), what is \(\vec{x}\) ?
Answer
(A)
\(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)
22
In a regular hexagon ABCDEF, if \(\vec{OA} = \vec{a}\) and \(\vec{OB} = \vec{b}\), which of the following represents \(\vec{OD}\)?
Answer
(C)
\(- \vec{a} - \vec{b}\)
23
In a column vector \[\begin{bmatrix} 3 \\ -2 \end{bmatrix}\], what does the 3 represent?
Answer
(B)
Movement to the right.
24
Which of the following is the correct way to find the vector that represents going from point A to point B, given their position vectors?
Answer
(A)
Subtract the position vector of A from the position vector of B.
25
In a triangle, let \(\vec{AB} = \vec{p}\) and \(\vec{BC} = \vec{q}\). What is \(\vec{CA}\)?
Answer
(B)
\(-\vec{p} - \vec{q}\)
26
Given \(p = \begin{pmatrix} 1 \\ 4 \end{pmatrix}\), which of the following is -2p?
Answer
(A)
\(\begin{pmatrix} -2 \\ -8 \end{pmatrix}\)
27
Which of the following can be found using a vector?
Answer
A
B
C
28
Given a = \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\), b = \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\), and c = \(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\), what is the value of a - b + c?
Answer
(B)
\(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\)
29
Given \(\vec{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\), which of the following represents the magnitude of \( \vec{a} + \vec{b} \)?
Answer
(B)
1
30
If the position vectors of points A and B are \(\vec{a}\) and \(\vec{b}\) respectively, which of the following represents the vector \(\vec{BA}\)?
Answer
(C)
\(\vec{a} - \vec{b}\)
31
Given the points A(1, 2) and B(4, 6), what is the column vector \(\vec{AB}\)?
Answer
(A)
\(\begin{bmatrix} 3 \\ 4 \end{bmatrix}\)
32
Given \(\vec{a} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}\), what is |\(\vec{a} - \vec{b}\)|?
Answer
(B)
5
33
If a vector \(\vec{a} \) is represented as \[\begin{bmatrix} 1 \\ 3 \end{bmatrix}\], and vector \(\vec{b} \) is represented as \[\begin{bmatrix} 2 \\ -1 \end{bmatrix}\], what is \( \vec{a} + \vec{b} \) ?
Answer
(A)
\(\begin{bmatrix} 3 \\ 2 \end{bmatrix}\)
34
If the position vector of point A is \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\), and the position vector of point B is \(\begin{pmatrix} 1 \\ 5 \end{pmatrix}\), then \(\vec{AB}\) is:
Answer
(B)
\(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\)
35
If \(\vec{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}\), what is 2\(\vec{a}\) - \(\vec{b}\) ?
Answer
(C)
\(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\)
36
In the parallelogram ABCD, which of the following is correct?
Answer
(B)
\(\vec{AB} + \vec{BC} = \vec{AC}\)
37
If \( ec{OA} = egin{pmatrix} 3 \\ 1 \end{pmatrix}\) and M is the midpoint of OA, then what is \(\vec{OM}\)?
Answer
(A)
\(\begin{pmatrix} 1.5 \\ 0.5 \end{pmatrix}\)
38
In a parallelogram OABC, which of the following must be true?
Answer
A
C
D
39
If \(p = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) and \(q = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\), find the magnitude of p - q.
Answer
(B)
\(\sqrt{5}\)
40
Given that \(\vec{p} = 2\vec{a}\) and \(\vec{q} = \vec{a} + \vec{b}\), which of the following correctly describes \(\vec{p} - \vec{q}\)?
