## Exercise Set 1A

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This set of questions will primarily test a student’s knowledge of the following lessons:

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**1.1 – Scalar and Vectors**

**1.2 – Visual Representation of Vectors**

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We recommend that **students of all levels** attempt the set below by following the suggested approach as stated within that set.

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For each question, we recommend that each student uses the following approach until they feel confident that they fully understand the answer:

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**Answer \(\rightarrow\) Solution \(\rightarrow\) Walkthrough \(\rightarrow\) Teacher Chat**

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**Exercise Set 1A**

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These questions are considered **beginner to intermediate Ordinary Level** questions.

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We recommend that students move on to the next lesson only once they have answered \(\mathbf{6}\)** questions fully correct **within this set.

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If a student reaches the end of this set without getting \(6\) questions fully correct, we recommend that they first quickly **review the lessons stated above** before moving on to the next lesson.

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If a student still feels doubtful that they are fully prepared for these questions if they were to appear on their exam, we suggest that they **book a grind** with Mr. Kenny.

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## Question 1

Consider the following vectors.

Which of these vectors

**(a)** are directed west?

**(b)** are directed east?

**(c)** has the largest magnitude?

**(d)** has the smallest magnitude?

**(a)** \(\vec{c}\) only

**(b)** \(\vec{a}\) and \(\vec{b}\)

**(c)** \(\vec{c}\)

**(d)** \(\vec{b}\)

**(a)** \(\vec{c}\) only

**(b)** \(\vec{a}\) and \(\vec{b}\)

**(c)** \(\vec{c}\)

**(d)** \(\vec{b}\)

**(a)** As the arrow representing \(\vec{c}\) is pointing to the left, that vector is directed west.

**(b)** As the arrows representing \(\vec{a}\) and \(\vec{b}\) are pointing to the right, both of these vectors are directed east.

**(c)** As the arrow representing \(\vec{c}\) is the longest, \(\vec{c}\) has the largest magnitude of the three.

**(d)** As the arrow representing \(\vec{b}\) is the shortest, \(\vec{b}\) has the smallest magnitude of the three.

**Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.**

**In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!**

## Question 2

Vector \(\vec{a}\) has a magnitude of \(12\) metres and a direction east.

State the magnitude of this vector:

**(a)** in centimetres

**(b)** in millimetres

**(c)** in kilometres

**(a) **\(1{,}200\mbox{ cm}\)

**(b) **\(12{,}000\mbox{ mm}\)

**(c) **\(0.012\mbox{ km}\)

**(a)**

\begin{align}12\times 100=1{,}200\mbox{ cm}\end{align}

**(b)**

\begin{align}12\times 1{,}000=12{,}000\mbox{ mm}\end{align}

**(c)**

\begin{align}12\times \frac{1}{1{,}000}=0.012\mbox{ km}\end{align}

**(a)** As there are \(100\) centimetres in a metre, there are \(12\times 100=1{,}200\) centimetres in \(12\) metres.

**(b)** As there are \(1{,}000\) millimetres in a metre, there are \(12\times 1{,}000=12{,}000\) millimetres in \(12\) metres.

**(c)** As there are \(1{,}000\) metres in a kilometre, there are \(\dfrac{1}{1{,}000}=0.001\) kilometres in a metre.

Therefore, there are \(12\times 0.001=0.012\) kilometres in \(12\) metres.

**Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.**

**In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!**

## Question 3

Consider the following vector \(\vec{a}\).

Using a ruler:

**(a)** Redraw \(\vec{a}\).

**(b)** Construct a vector \(\vec{b}\) whose magnitude is half that of \(\vec{a}\) and that points in the same direction as \(\vec{a}\).

**(c)** Construct a vector \(\vec{c}\) whose magnitude is the same as \(\vec{a}\) and that points in the opposite direction to \(\vec{a}\).

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)** This vector has been redrawn using a ruler as below.

**(b)** As the magnitude of \(\vec{b}\) is half that of \(\vec{a}\), the arrow representing \(\vec{b}\) will be half as long as the arrow representing \(\vec{a}\).

As \(\vec{b}\) points in the same direction as \(\vec{a}\), the arrow representing \(\vec{b}\) points in the same direction as the arrow representing \(\vec{a}\).

**(c)** As the magnitude of \(\vec{c}\) is the same as \(\vec{a}\), the arrow representing \(\vec{c}\) will be the same length as the arrow representing \(\vec{a}\).

As \(\vec{c}\) points in the opposite direction to \(\vec{a}\), the arrow representing \(\vec{b}\) points in the opposite direction to the arrow representing \(\vec{a}\).

**Video Walkthroughs are currently in the process of being gradually uploaded, with that process expected to be finished by the end of October.**

**In the meantime, if you don't understand something in the Walkthrough above, feel free to reach out to Mr. Kenny on Teacher Chat!**

## Question 4

The speed limit of most Irish motorways is \(120\) kilometres per hour.

Using a calculator, state this speed limit in metres per second.

\(33\dfrac{1}{3}\mbox{ m/s}\)

\begin{align}120\mbox{ km/h}&= \frac{120 \mbox{ kilometres}}{1\mbox{ hour}}\\&=\frac{120\times 1{,}000\mbox{ metres}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{120{,}000}{3{,}600}\frac{\mbox{metres}}{\mbox{second}}\\&=33\frac{1}{3}\mbox{ m/s}\end{align}

There are \(1{,}000\) metres in a kilometre and there are \(60\times 60=3{,}600\) seconds in an hour.

Therefore, we can rewrite this speed limit as follows:

\begin{align}120\mbox{ km/h}&= \frac{120 \mbox{ kilometres}}{1\mbox{ hour}}\\&=\frac{120\times 1{,}000\mbox{ metres}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{120{,}000}{3{,}600}\frac{\mbox{metres}}{\mbox{second}}\\&=33\frac{1}{3}\mbox{ m/s}\end{align}

where the final answer was found using a calculator.

## Question 5

**(a)** Construct any vector \(\vec{a}\) that is pointing east.

**(b)** Construct a vector \(\vec{b}\) that is pointing west and has twice the magnitude of \(\vec{a}\).

**(a)**

**(b)**

**(a)**

**(b)**

**(a)** We are being asked to construct a vector \(\vec{a}\) of any magnitude whose direction is east.

One such vector is as shown below.

**(b)** As the magnitude of \(\vec{b}\) is twice that of \(\vec{a}\), the arrow representing \(\vec{b}\) will be twice as long as the arrow representing \(\vec{a}\).

As \(\vec{b}\) points in the opposite direction to \(\vec{a}\), the arrow representing \(\vec{b}\) points in the opposite direction to the arrow representing \(\vec{a}\).

## Question 6

The length of a typical ruler is approximately \(30\mbox{ cm}\).

State the length of such a ruler:

**(a)** in millimetres

**(b)** in metres

**(a) **\(300\mbox{ mm}\)

**(b) **\(0.3\mbox{ m}\)

**(a)**

\begin{align}30\times \dfrac{1{,}000}{100}=300\mbox{ mm}\end{align}

**(b)**

\begin{align}30\times \dfrac{1}{100}=0.3\mbox{ m}\end{align}

**(a)** There are \(100\) centimetres in a metre and there are \(1{,}000\) millimetres in a metre.

Therefore, there are \(\dfrac{1{,}000}{100}=10\) millimetres in a centimetre.

Hence, a ruler is \(30\times 10=300\) millimetres long.

**(b)** As there are \(100\) centimetres in a metre, there are \(\dfrac{1}{100}=0.01\) metres in a centimetre.

Therefore, a ruler is \(30\times 0.01=0.3\) metres long.

## Question 7

Consider the following vector \(\vec{a}\).

**(a)** State the direction of this vector.

**(b)** Construct any two vectors pointing in the opposite direction to \(\vec{a}\).

**(a)** West (or to the left).

**(b)**

**(a)** West (or to the left).

**(b)**

**(a)** This vector is pointing west (or to the left).

**(b)** We are being asked to construct two vectors of any magnitude that are pointing east (or to the right), two of which are shown below.

## Question 8

A train is travelling at a constand speed of \(30\mbox{ m/s}\).

Using a calculator, write this train’s speed in \(\mbox{km/h}\).

\(108\mbox{ km/h}\)

\begin{align}30\mbox{ m/s}&= \frac{30 \mbox{ metres}}{1\mbox{ second}}\\&=\frac{30\times \frac{1}{1{,}000}\mbox{ kilometres}}{1\times \frac{1}{3{,}600}\mbox { hours}}\\&=\frac{\frac{30}{1{,}000}}{\frac{1}{3{,}600}}\frac{\mbox{kilometres}}{\mbox{hour}}\\&=108\mbox{ km/h}\end{align}

There are \(1{,}000\) metres in a kilometre. Therefore, there are \(\dfrac{1}{1{,}000}\) kilometres in a metre.

There are \(60\times 60=3{,}600\) seconds in an hour. Therefore, there are \(\dfrac{1}{3{,}600}\) hours in a second.

Hence, we can rewrite this train’s speed as follows:

\begin{align}30\mbox{ m/s}&= \frac{30 \mbox{ metres}}{1\mbox{ second}}\\&=\frac{30\times \frac{1}{1{,}000}\mbox{ kilometres}}{1\times \frac{1}{3{,}600}\mbox { hours}}\\&=\frac{\frac{30}{1{,}000}}{\frac{1}{3{,}600}}\frac{\mbox{kilometres}}{\mbox{hour}}\\&=108\mbox{ km/h}\end{align}

where the final answer was found using a calculator.

## Question 9

Consider the following vector \(\vec{w}\).

**(a)** Redraw \(\vec{w}\).

**(b)** Construct a vector \(\vec{x}\) whose magnitude is half that of \(\vec{w}\) and that points in the opposite direction to \(\vec{w}\).

**(c)** Construct a vector \(\vec{y}\) whose magnitude is four times that of \(\vec{x}\) and that points in the opposite direction to \(\vec{x}\).

**(d)** What is the relationship between \(\vec{w}\) and \(\vec{y}\)?

**(a)**

**(b)**

**(c)**

**(d)** \(\vec{y}\) has twice the magnitude of \(\vec{w}\) and both have the same direction.

**(a)**

**(b)**

**(c)**

**(d)** \(\vec{y}\) has twice the magnitude of \(\vec{w}\) and both have the same direction.

**(a)** This vector has been redrawn as below.

**(b)** As the magnitude of \(\vec{x}\) is half that of \(\vec{w}\), the arrow representing \(\vec{x}\) will be half as long as the arrow representing \(\vec{w}\).

As \(\vec{x}\) points in the opposite direction to \(\vec{w}\), the arrow representing \(\vec{x}\) points in the opposite direction to the arrow representing \(\vec{w}\).

**(c)** As the magnitude of \(\vec{y}\) is four times that of \(\vec{x}\), the arrow representing \(\vec{y}\) will be four times as long as the arrow representing \(\vec{x}\).

As \(\vec{y}\) points in the opposite direction to \(\vec{x}\), the arrow representing \(\vec{y}\) points in the opposite direction to the arrow representing \(\vec{x}\).

**(d)** As \(\vec{x}\) has half the magnitude of \(\vec{w}\), and as \(\vec{y}\) has four times the magnitude of \(\vec{x}\), \(\vec{y}\) has \(\dfrac{1}{2}\times 4=2\) times the magnitude of \(\vec{w}\).

Also, according to the above diagrams, \(\vec{y}\) is pointing in the same direction as \(\vec{w}\).

## Question 10

Cement is being used to construct a house at rate of \(9\) kilograms per hour.

State this rate in grams per second.

\(2.5\mbox{ g/s}\)

\begin{align}9\mbox{ kg/h}&= \frac{9 \mbox{ kilograms}}{1\mbox{ hour}}\\&=\frac{9\times 1{,}000\mbox{ grams}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{9{,}000}{3{,}600}\frac{\mbox{grams}}{\mbox{second}}\\&=2.5\mbox{ g/s}\end{align}

There are \(1{,}000\) grams in a kilogram and there are \(60\times 60=3{,}600\) seconds in an hour.

Therefore, we can rewrite this rate as follows:

\begin{align}9\mbox{ kg/h}&= \frac{9 \mbox{ kilograms}}{1\mbox{ hour}}\\&=\frac{9\times 1{,}000\mbox{ grams}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{9{,}000}{3{,}600}\frac{\mbox{grams}}{\mbox{second}}\\&=2.5\mbox{ g/s}\end{align}

where the final answer was found using a calculator.