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 \(\vec{AB}\).

  回答

  (C)

  \(\begin{pmatrix} -1 \\ -2 \end{pmatrix}\)
• 2

  Which of the following is the correct method for finding the magnitude of the vector \(\begin{pmatrix} x \\ y \end{pmatrix}\)?

  回答

  (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 \(2\vec{a} - \vec{b}\)?

  回答

  (B)

  \(\begin{pmatrix} 5 \\ 5 \end{pmatrix}\)
• 4

  What does the top number in a column vector indicate?

  回答

  (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 \(\vec{q}\).

  回答

  (B)

  \(\vec{p} + \vec{q}\)
• 6

  If \(\vec{p} = \begin{pmatrix} 4 \\ -2 \end{pmatrix}\), what is - \(\frac{1}{2}\vec{p}\)?

  回答

  (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\(\vec{b}\)?

  回答

  (A)

  3\(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) + 2\(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\)
• 8

  Given \(\vec{u} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\) and \(\vec{w} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}\), calculate \(2\vec{u} + \vec{w}\).

  回答

  (C)

  \(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\)
• 9

  The notation \[\begin{bmatrix} x \\ y \end{bmatrix}\] is often used to represent:

  回答

  (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\(\vec{a}\) + \(\vec{b}\)?

  回答

  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?

  回答

  (B)

  \(\begin{bmatrix} -4 \\ 6 \end{bmatrix}\)
• 12

  If \(\vec{a} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\), what is the magnitude of 2\(\vec{a}\)?

  回答

  (B)

  \(2\sqrt{2}\)
• 13

  In a triangle, \(\vec{AB} = \vec{p}\) and \(\vec{BC} = \vec{q}\). What is the value of \(\vec{AC}\)?

  回答

  (B)

  \(\vec{p} + \vec{q}\)
• 14

  Which of the following operations is NOT defined for vectors?

  回答

  (D)

  Vector division
• 15

  What is the magnitude of the vector \(\begin{pmatrix} 5 \\ -12 \end{pmatrix}\)?

  回答

  (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 \( 3\vec{u} + 2\vec{v} \)?

  回答

  (C)

  \(\begin{bmatrix} 1 \\ 10 \end{bmatrix}\)
• 17

  If \(\vec{u} = 4\vec{v}\), which statement is ALWAYS true?

  回答

  (B)

  \(\vec{u}\) and \(\vec{v}\) have the same direction.
• 18

  If \(\vec{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\), what is -3\(\vec{a}\)?

  回答

  (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?

  回答

  (C)

  \(\begin{bmatrix} 5 \\ 7 \end{bmatrix}\)
• 20

  Given the points A(1, -1) and B(3, 2), the vector \(\vec{AB}\) is:

  回答

  (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 \(\vec{AC}\)?

  回答

  (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:

  回答

  (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 \(\vec{a} + \vec{b}\)?

  回答

  (B)

  \(\sqrt{10}\)
• 24

  If \( \vec{a} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \), what is \( -2\vec{a} \) equal to?

  回答

  (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 \(\vec{BC}\)?

  回答

  (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?

  回答

  (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 \(\vec{a} + 2\vec{b}\)?

  回答

  (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?

  回答

  (A)

  \(\vec{0}\)
• 29

  If \(\vec{p} + \vec{q} = \vec{0}\), what is the relationship between \(\vec{p}\) and \(\vec{q}\)?

  回答

  (B)

  They are equal in magnitude and opposite in direction.
• 30

  In the context of column vectors, what does the term 'scalar' represent?

  回答

  (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?

  回答

  (A)

  [4, -2]
• 32

  If two vectors are parallel, what does that tell you about their directions?

  回答

  (C)

  They have the same direction or opposite directions
• 33

  What does the top number in a column vector represent?

  回答

  (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?

  回答

  (D)

  All of the above are possible
• 35

  If a shape is translated by the vector [4, -1], the shape moves:

  回答

  (B)

  4 units right and 1 unit down
• 36

  If two vectors are parallel, what can be said about the relationship of their components?

  回答

  (B)

  Their components are proportional.
• 37

  What type of quantity is the magnitude of a vector?

  回答

  (B)

  A scalar
• 38

  What is the result of adding a vector to its negative?

  回答

  (C)

  The zero vector.
• 39

  Which of the following is used for vector addition?

  回答

  (A)

  Adding the x-components and y-components separately
• 40

  If AB = [4, -2] and AC = [1, 3], what is the vector CB?

  回答

  (B)

  [-3, 5]
• 41

  What is the term used to describe a vector whose direction can be reversed?

  回答

  (B)

  Opposite vector
• 42

  If a = [1, 2] and b = [-3, 1], what is the magnitude of the vector 2a + b?

  回答

  (C)

  5
• 43

  If vector AB is [3, 2], and we want to find the vector BA, what is the new vector?

  回答

  (C)

  [-3, -2]
• 44

  What can you conclude about the vectors AB and CD if AB = CD?

  回答

  (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?

  回答

  (A)

  [4, 1]
• 46

  If the position vectors of points P and Q are p and q respectively, what represents the vector QP?

  回答

  (C)

  p - q
• 47

  Which of the following statements is FALSE regarding vector notation?

  回答

  (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?

  回答

  (B)

  a + b
• 49

  If a vector 'v' is [1, 0], and is multiplied by 0, what is the resulting vector?

  回答

  (C)

  [0, 0]
• 50

  What does the term 'resultant vector' refer to?

  回答

  (B)

  The vector sum of two or more vectors.