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It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Past Exam Questions and Solutions on Vectors

Calculators: Calculators
Prerequisites:
(1.) Expressions and Equations
(2.) Trigonometry

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.

For JAMB Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.
Unless specified otherwise, any question labeled JAMB is a question from JAMB Physics

For WASSCE Students: Unless specified otherwise:
Any question labeled WASCCE is a question from WASCCE Physics
Any question labeled WASSCE-FM is a question from the WASSCE Further Mathematics/Elective Mathematics

For NYSED Students
Unless specified otherwise, any question labeled NYSED is a question from NYSED Physics

For GCSE Students
All work is shown to satisfy (and actually exceed) the minimum for awarding method marks.
Calculators are allowed for some questions. Calculators are not allowed for some questions.

For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.

Solve all questions.
Show all work.

(1.) JAMB Which of the following is NOT a vector quantity?
A. Force
B. Altitude
C. Weight
D. Displacement
E. Acceleration

The correct answer is Option B. Altitude
Please review the reasons: Examples of Vectors and Scalars with Explanations
(2.) ACT The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north.
Maria is jogging south at 12 miles per hour.
One of the following vectors represents Maria's velocity, in miles per hour.
Which one?

$A.\;\; -12i \\[3ex] B.\;\; -12j \\[3ex] C.\;\; 12i \\[3ex] D.\;\; 12j \\[3ex] E.\;\; 12i + 12j \\[3ex]$

Let us represent the information on a graph using the four main compass directions

$j = 1\;mile/hour\;\;north \\[3ex] -j = 1\;mile/hour\;\;south \\[3ex] -12j = 12\;miles/hour\;\;south$
(3.)

(4.) ACT Given that u and v are vectors such that u = $\langle -1, 3 \rangle$ and v = $\langle 5, 8 \rangle$, what is the component form of the vector u + v?

$F.\;\; \langle 2, 13 \rangle \\[3ex] G.\;\; \langle 4, 5 \rangle \\[3ex] H.\;\; \langle 4, 11 \rangle \\[3ex] J.\;\; \langle 6, 5 \rangle \\[3ex] K.\;\; \langle 6, 11 \rangle \\[3ex]$

$u + v \\[3ex] = \langle -1, 3 \rangle + \langle 5, 8 \rangle \\[3ex] = \langle -1 + 5, 3 + 8 \rangle \\[3ex] = \langle 4, 11 \rangle$
(5.)

(6.) NYSED Regents Examination Which pair of quantities represent scalar quantities?
(1) displacement and velocity
(2) displacement and time
(3) energy and velocity
(4) energy and time

The correct answer is Option (4.) energy and time
Please review the reasons: Examples of Vectors and Scalars with Explanations
(7.)

(8.) ACT When the vector ai + 3j is added to the vector -2i + bj, the sum is 6i - 6j
What are the values of a and b?
A. a = -9 and b = 8
B. a = -8 and b = 9
C. a = -4 and b = 3
D. a = 4 and b = -3
E. a = 8 and b = -9

$(ai + 3j) + (-2i + bj) = 6i - 6j \\[3ex] \implies \\[3ex] ai + -2i = 6i \\[3ex] ai - 2i = 6i \\[3ex] i(a - 2) = 6i \\[3ex] a - 2 = 6 \\[3ex] a = 6 + 2 \\[3ex] a = 8 \\[5ex] 3j + bj = -6j \\[3ex] j(3 + b) = -6j \\[3ex] 3 + b = -6 \\[3ex] b = -6 - 3 \\[3ex] b = -9 \\[3ex] a = 8 \;\;and\;\; b = -9$
(9.) WASSCE-FM A(-3, 1), B(1, 2), C(0, -1) and D(x, y) are the vertices of a parallelogram.
Using vector method, determine the coordinates of D.

Let us draw the parallelogram so it can help us determine the position of D

$A(-3, 1) \\[3ex] D(x, y) \\[3ex] B(1, 2) \\[3ex] C(0, -1) \\[3ex] \overrightarrow{DA} = \overrightarrow{CB}...opposite\;\;sides\;\;of\;\;a\;\;parallelogram\;\;are\;\;equal \\[3ex] \langle x - (-3), y - 1 \rangle = \langle 0 - 1, -1 - 2 \rangle \\[3ex] \langle x + 3, y - 1 \rangle = \langle -1, -3 \rangle \\[3ex] \implies \\[3ex] x + 3 = -1 \\[3ex] x = -1 - 3 \\[3ex] x = -4 \\[3ex] y - 1 = -3 \\[3ex] y = -3 + 1 \\[3ex] y = -2 \\[3ex] \therefore D(x, y) = D(-4, -2)$
(10.) NYSED Regents Examination Which combination correctly pairs a vector quantity with its corresponding unit?
(1) weight and kg
(2) velocity and m/s
(3) speed and m/s
(4) acceleration and m²/s

The vectors are: weight, velocity and acceleration
Please review the reasons: Examples of Vectors and Scalars with Explanations

The unit of weight is Newton (N)
The unit of velocity is m/s
The unit of acceleration is m/s²
Therefore, the correct answer is Option (2.) velocity and m/s
(11.)

(12.) GCSE Here is vector a.

Circle the column vector that represents a

$\begin{pmatrix} 3 \\[3ex] 2 \end{pmatrix} \;\;\;\;\;\;\; \begin{pmatrix} -3 \\[3ex] 2 \end{pmatrix} \;\;\;\;\;\;\; \begin{pmatrix} 3 \\[3ex] -2 \end{pmatrix} \;\;\;\;\;\;\; \begin{pmatrix} -3 \\[3ex] -2 \end{pmatrix} \\[7ex]$

$Initial\;\;point = (1, 4) \\[3ex] Terminal\;\;point = (4, 2) \\[3ex] a: (1, 4) \rightarrow (4, 2) \\[3ex] a: \langle 4 - 1, 2 - 4 \rangle \\[3ex] a \langle 3, -2 \rangle \\[3ex] Column\;\;vector = \begin{pmatrix} 3 \\[3ex] -2 \end{pmatrix}$
(13.) CSEC Three points, O, P and R, are shown on the grid below.
O is the origin.

(i) Write the position vector of R, $\overrightarrow{OR}$, in the form, $\begin{pmatrix} a \\[3ex] b \end{pmatrix} \\[5ex]$ (ii) Another point, Q, is located in such a way that $\overrightarrow{QR} = \begin{pmatrix} 2 \\[3ex] -4 \end{pmatrix} \\[5ex]$ Using this information, plot the point Q on the graph.

(iii) Determine $|\overrightarrow{QR}|$, the magnitude of $\overrightarrow{QR}$

(iv) Show, by calculation, that OPQR is a parallelogram.

$(i) \\[3ex] R(5, 1)...from\;\;the\;\;graph \\[3ex] \overrightarrow{OR} = \begin{pmatrix} 5 \\[3ex] 1 \end{pmatrix} \\[5ex] (ii) \\[3ex] Let\;\; \vec{Q} = \begin{pmatrix} x \\[3ex] y \end{pmatrix} \\[5ex] \vec{R} = \begin{pmatrix} 5 \\[3ex] 1 \end{pmatrix} \\[5ex] \overrightarrow{QR} = \vec{R} - \vec{Q} \\[3ex] \begin{pmatrix} 2 \\[3ex] -4 \end{pmatrix} = \begin{pmatrix} 5 \\[3ex] 1 \end{pmatrix} - \begin{pmatrix} x \\[3ex] y \end{pmatrix} \\[5ex] \implies \\[3ex] 2 = 5 - x \\[3ex] x = 5 - 2 \\[3ex] x = 3 \\[3ex] -4 = 1 - y \\[3ex] y = 1 + 4 \\[3ex] y = 5 \\[3ex] Q(x, y) = Q(3, 5) \\[3ex]$ The plot of point Q on the graph is:

$(iii) \\[3ex] \overrightarrow{QR} = \langle 2, -4 \rangle \\[3ex] |\overrightarrow{QR}| = \sqrt{2^2 + (-4)^2} \\[3ex] = \sqrt{4 + 16} \\[3ex] = \sqrt{20} \\[3ex] = \sqrt{4 * 5} \\[3ex] = \sqrt{4} * \sqrt{5} \\[3ex] = 2\sqrt{5} \\[3ex] (iv) \\[3ex] OPQR \\[3ex] O(0, 0) \\[3ex] P(-2, 4) ...from\;\;the\;\;graph \\[3ex] Q(3, 5) \\[3ex] R(5, 1) \\[3ex]$ Let us represent these points on the graph

Opposite sides of a parallelogram are equal

$\overrightarrow{OP} = \langle 0 - (-2), 0 - 4 \rangle \\[3ex] \overrightarrow{OP} = \langle 0 + 2, 0 - 4 \rangle \\[3ex] \overrightarrow{OP} = \langle 2, -4 \rangle \\[3ex] \overrightarrow{RQ} = \langle 5 - 3, 1 - 5 \rangle \\[3ex] \overrightarrow{RQ} = \langle 2, -4 \rangle \\[3ex] \overrightarrow{OP} = \overrightarrow{RQ} \\[3ex] Also: \\[3ex] \overrightarrow{PQ} = \langle 3 - (-2), 5 - 4 \rangle \\[3ex] \overrightarrow{PQ} = \langle 3 + 2, 5 - 4 \rangle \\[3ex] \overrightarrow{PQ} = \langle 5, 1 \rangle \\[3ex] \overrightarrow{OR} = \langle 5 - 0, 1 - 0 \rangle \\[3ex] \overrightarrow{OR} = \langle 5, 1 \rangle \\[3ex] \overrightarrow{PQ} = \overrightarrow{OR} \\[3ex]$ The opposite sides are equal.
OPQR is a parallelogram.
(14.) ACT Two vectors are shown in the standard (x, y) coordinate plane below.

One of the following vectors in the standard (x, y) coordinate plane is the sum of these 2 vectors.
Which one?

We can solve this question using at least 2 approaches: Graphically and Algebraically
The graphical approach is recommended especially in this case and because the ACT is a timed test.
We shall do it graphically first and then algebraically

Let the first vector be $\vec{A}$
Let the second vector be $\vec{B}$
Construction: Join the initial point of $\vec{B}$ to the terminal point of $\vec{A}$

Graphically:

$\vec{C} = \vec{A} + \vec{B}$

Looking at all the options, the correct option is Option E.

Algebraically:
Express vectors $\vec{A}$ and $\vec{B}$ in component form
Because their initial points are the origin, their components forms are their respective terminal points

$\vec{A} \langle 1, 4 \rangle \\[3ex] \vec{B} \langle 3, 2 \rangle \\[3ex] \vec{A} + \vec{B} \\[3ex] = \langle 1 + 3, 4 + 2 \rangle \\[3ex] = \langle 4, 6 \rangle \\[3ex]$ The correct option whose terminal point is $\langle 4, 6 \rangle$ is Option E.
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(19.) NYSED The scaled diagram below represents two forces acting concurrently at point P.
The magnitude of force A is 32 newtons and the magnitude of force B is 20. newtons.
The angle between the directions of force A and force B is 120°

(19.1) Determine the linear scale used in the diagram.
(19.2) On the diagram in your answer booklet, use a protractor and a ruler to construct a scaled vector to represent the resultant of forces A and B. Label the vector R.
(19.3) Determine the magnitude of the resultant force.

Construction:
(1.) Draw a horizontal line from Point P
90° is formed
The other angle = 120° - 90° = 30°

(2.) Draw a similar horizontal line from point A
(3.) Draw the smilar force B such that it is 30° from the horizontal line drawn from point A
(This means that the angle that force B makes with force A to the horizontal is 30°)
(4.) The other angle (angle that force B makes with force A to the vertical) = 90° - 30° = 60°

Let blue color = resultant force

$\underline{\triangle ABR} \\[3ex] \vec{R}^2 = \vec{A}^2 + \vec{B}^2 - 2 * \vec{A} * \vec{B} \cos 60^\circ \\[3ex] \vec{R}^2 = 32^2 + 20^2 - 2(32)(20) \cos 60 \\[3ex] \vec{R}^2 = 1024 + 400 - 1280(0.5) \\[3ex] \vec{R}^2 = 1424 - 640 \\[3ex] \vec{R}^2 = 784 \\[3ex] \vec{R} = \sqrt{784} \\[3ex] \vec{R} = 28\;newtons$
(20.) ACT Representatives of vectors u, v, p, q, and r are shown in the standard (x, y) coordinate plane below.

One of the following vectors is equal to the vector u + v.
Which one?

$F.\;\; -r \\[3ex] G.\;\; -q \\[3ex] H.\;\, -p \\[3ex] J.\;\; p \\[3ex] K.\;\; q \\[3ex]$

Let us express these vectors in component form
Then, we add the vectors u + v

$\underline{vector\;\;u} \\[3ex] initial\;\;point = (0, 1) \\[3ex] terminal\;\;point = (1, 3) \\[3ex] u: (0, 1) \rightarrow (1, 3) \\[3ex] u: \langle 1 - 0, 3 - 1 \rangle \\[3ex] u \langle 1, 2 \rangle \\[3ex] \underline{vector\;\;v} \\[3ex] initial\;\;point = (2, 0) \\[3ex] terminal\;\;point = (3, -1) \\[3ex] v: (2, 0) \rightarrow (3, -1) \\[3ex] v: \langle 3 - 2, -1 - 0 \rangle \\[3ex] v \langle 1, -1 \rangle \\[3ex] \underline{Addition\;\;of\;\;u\;\;and\;\;v} \\[3ex] u + v \\[3ex] = \langle 1, 2 \rangle + \langle 1, -1 \rangle \\[3ex] = \langle 1 + 1, 2 + -1 \rangle \\[3ex] = \langle 2, 1 \rangle \\[3ex] \underline{vector\;\;p} \\[3ex] initial\;\;point = (-2, 0) \\[3ex] terminal\;\;point = (-2, 3) \\[3ex] p: (-2, 0) \rightarrow (-2, 3) \\[3ex] p: \langle -2 - (-2), 3 - 0 \rangle \\[3ex] p: \langle -2 + 2, 3 \rangle \\[3ex] p \langle 0, 3 \rangle \\[3ex] \underline{vector\;\;q} \\[3ex] initial\;\;point = (-3, -2) \\[3ex] terminal\;\;point = (-1, -1) \\[3ex] q: (-3, -2) \rightarrow (-1, -1) \\[3ex] q: \langle -1 - (-3), -1 - (-2) \rangle \\[3ex] q: \langle -1 + 3, -1 + 2 \rangle \\[3ex] q \langle 2, 1 \rangle \\[3ex] \underline{vector\;\;r} \\[3ex] initial\;\;point = (1, -2) \\[3ex] terminal\;\;point = (1, -1) \\[3ex] r: (1, -2) \rightarrow (1, -1) \\[3ex] r: \langle 1 - 1, -1 - (-2) \rangle \\[3ex] r: \langle 0, -1 + 2 \rangle \\[3ex] r \langle 0, 1 \rangle \\[3ex]$ The single vector that is equivalent to u + v is vector q

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(26.) NYSED A unit used for a vector quantity is
(1) watt           (3) kilogram
(2) newton          (4) second

watt is the unit for power
newton is the unit for force
kilogram is the unit for mass
second is the unit for time

So, which of those quantities is a vector?
Force is a vector. The unit of force is Newton (N)

Please review the reasons: Examples of Vectors and Scalars with Explanations
(27.)

(28.) ACT What vector is the result of adding the vectors $\langle 1, -2 \rangle$, $\langle 2, 3 \rangle$, and $\langle -4, 2 \rangle$ ?

$A.\;\; \langle -8, -12 \rangle \\[3ex] B.\;\; \langle -1, 3 \rangle \\[3ex] C.\;\; \langle 3, -3 \rangle \\[3ex] D.\;\; \langle 7, -1 \rangle \\[3ex] E.\;\; \langle 7, 7 \rangle \\[3ex]$

$\langle 1, -2 \rangle + \langle 2, 3 \rangle + \langle -4, 2 \rangle \\[3ex] \langle 1 + 2 + (-4), -2 + 3 + 2 \rangle \\[3ex] \langle -1, 3 \rangle$
(29.)

(30.) NYSED A displacement vector with a magnitude of 20. meters could have perpendicular components with magnitudes of
(1) 10. m and 10. m         (3) 12 m and 16 m
(2) 12 m and 8.0 m         (4) 16 m and 8.0 m

The perpendicular components are the vertical component and the horizontal component
The displacement vector can be seen as the hypotenuse of a right-angled triangle because the question asked us to determine the perpendicular components
So, let us begin with the simplest Pythagorean Triple: 3 - 4 - 5
5 is the hypotenuse
5 * ? = 20
5 * 4 = 20
Therefore the other legs are:
3 * 4 = 12
4 * 4 = 16
The Pythagorean triple dealing with 20 as the hypotenuse is:
12 - 16 - 20
Therefore, the perpendicular components are: 12 m and 16 m
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(41.) ACT The vector i represents 1 mile per hour east, and the vector j represents 1 mile per hour north.
According to her GPS, at a particular instant, Tia is biking 30° west of north at 16 miles per hour.
One of the following vectors represents Tia's velocity, in miles per hour, at that instant.
Which one?

$A.\;\; -8i - 8\sqrt{3}j \\[3ex] B.\;\; -8i + 8\sqrt{3}j \\[3ex] C.\;\; 8i + 8\sqrt{3}j \\[3ex] D.\;\; 8\sqrt{3}i - 8j \\[3ex] E.\;\; 8\sqrt{3}i + 8j \\[3ex]$

30° W of N = N 30° W (Bearings and Distances)
It will be useful to represent the information on a diagram

$SOHCAHTOA \\[3ex] \sin 30 = \dfrac{x}{16} \\[5ex] x = 16 \sin 30 \\[3ex] x = 16\left(\dfrac{1}{2}\right) \\[5ex] x = 8 \\[3ex] x = -8i...negative\;\;x-axis \\[3ex] \cos 30 = \dfrac{y}{16} \\[5ex] y = 16 \cos 30 \\[3ex] y = 16\left(\dfrac{\sqrt{3}}{2}\right) \\[5ex] y = 8\sqrt{3} \\[3ex] y = 8\sqrt{3}j ... positive\;\;y-axis \\[3ex] Tia's\;\;velocity\;\;in\;\;mph = -8i + 8\sqrt{3}j$
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