Section: 17 | Algebra |
Help Manual

Page of 1
Type a page number and hit Enter.
/1
Back to Search Results
Type a page number and hit Enter.
Summary of table differences
No records found.
How to Cite this Reference
 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# 2 ALGEBRA

The solutions of the equation ${ax}^{2}+bx+c=0$ , where $a\ne 0$ , are given by: $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ .

## 2.2 VECTOR ALGEBRA

### 2.2.1 Definitions

Any quantity which is completely determined by its magnitude is called a scalar. Examples include: mass, density, and temperature. Any quantity which is completely determined by its magnitude and direction is called a vector. Examples include: velocity, acceleration, force. A vector quantity is usually represented by a boldfaced letter such as $V$ . Two vectors ${V}_{1}$ and ${V}_{2}$ are equal to one another if they have equal magnitudes and are acting in the same directions. A negative vector, written as $-V$ , is one which acts in the opposite direction to $V$ , but is of equal magnitude to it. The magnitude of $V$ is written $|V|$ or simply $v$ . The unit vector$\frac{V}{|V|}$ (when $|V|\ne 0\right)$ is that vector which has the same direction as $V$ , but has a magnitude of unity (sometimes represented as $\stackrel{\wedge }{V}$ ).

The vector sum of ${V}_{1}$ and ${V}_{2}$ is represented by ${V}_{1}+{V}_{2}$ . The vector sum of ${V}_{1}$ and $-{V}_{2}$ , or the difference of the vector ${V}_{2}$ from ${V}_{1}$ is represented by ${V}_{1}-{V}_{2}$ .

If $r$ is a scalar, then $rV=Vr$ and this represents a vector $r$ times the magnitude of $V$ , in the same direction as $V$ if $r$ is positive, and in the opposite direction if $r$ is negative. If $r$ and $s$ are scalars and ${V}_{1}$ , ${V}_{2}$ , ${V}_{3}$ are vectors then the following rules of scalars and vectors hold:
$\begin{array}{cc}\hfill {V}_{1}+{V}_{2}& ={V}_{2}+{V}_{1}\hfill \\ \multicolumn{1}{c}{\left(r+s\right){V}_{1}}& =r{V}_{1}+s{V}_{1}\hfill \\ \multicolumn{1}{c}{r\left({V}_{1}+{V}_{2}\right)}& =r{V}_{1}+r{V}_{2}\hfill \\ \multicolumn{1}{c}{{V}_{1}+\left({V}_{2}+{V}_{3}\right)}& =\left({V}_{1}+{V}_{2}\right)+{V}_{3}={V}_{1}+{V}_{2}+{V}_{3}\hfill \\ \multicolumn{1}{c}{}\end{array}$
The vector $0$ is a vector of zero length.

### 2.2.2 Vectors in Space

1. A plane is described by two distinct vectors ${V}_{1}$ and ${V}_{2}$ . Should these vectors not intersect each other, then one can be displaced parallel to itself until they do. Any other vector $V$ lying in this plane is given by
$V=r{V}_{1}+s{V}_{2}$
2. A position vector specifies the position in space of a point relative to a fixed origin. If ${V}_{1}$ and ${V}_{2}$ are the position vectors of the points $A$ and $B$ , relative to the origin $O$ , then any point $P$ on the line AB has a position vector $V$ given by
$V=r{V}_{1}+\left(1-r\right){V}_{2}$
The scalar '' $r$ '' can be taken as the metric representation of $P$ since $r=0$ implies $P=B$ and $r=1$ implies $P=A$ . If the point $P$ divides the line AB in the ratio $r:s$ then
$V=\left(\frac{r}{r+s}\right){V}_{1}+\left(\frac{s}{r+s}\right){V}_{2}$
3. The vectors ${V}_{1}$ , ${V}_{2},{V}_{3},\dots ,{V}_{n}$ are said to be linearly dependent if there exist scalars ${r}_{1},{r}_{2},{r}_{3},\dots ,{r}_{n}$ , not all zero, such that
${r}_{1}{V}_{1}+{r}_{2}{V}_{2}+\dots +{r}_{n}{V}_{n}=0$
4. A vector $V$ is linearly dependent upon the set of vectors $\left\{{V}_{1},{V}_{2},{V}_{3},\dots ,{V}_{n}\right\}$ if
$V={r}_{1}{V}_{1}+{r}_{2}{V}_{2}+{r}_{3}{V}_{3}+\dots +{r}_{n}{V}_{n}$
5. Three vectors are linearly dependent if and only if they are co-planar.
6. All points in space can be uniquely determined by linear dependence upon three base vectors i.e., three vectors any one of which is linearly independent of the other two. The simplest set of base vectors are the unit vectors along the coordinate axes. These are usually designated by $i$ , $j$ , and $k$ .
7. If $V$ is a vector in space, and $a$ , $b$ , and $c$ are the respective magnitudes of the projections of the vector along the axes then
$V=ai+bj+ck\mathrm{ }\text{and} \mathrm{ }|V|=\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}$
and the direction cosines of $V$ are
$cos\alpha =a/v,\mathrm{ }cos\beta =b/v,\mathrm{ }cos\gamma =c/v.$
8. The law of vector addition yields
${V}_{1}+{V}_{2}=\left({a}_{1}+{a}_{2}\right)i+\left({b}_{1}+{b}_{2}\right)j+\left({c}_{1}+{c}_{2}\right)k$
9. ### 2.2.3 The Scalar, Dot, or Inner Product of Two Vectors

This product is represented as ${V}_{1}·{V}_{2}$ and is defined to be equal to ${v}_{1}{v}_{2}cos\theta$ , where ${v}_{1}=|{V}_{1}|,{v}_{2}=|{V}_{2}|$ , and $\theta$ is the angle from ${V}_{1}$ to ${V}_{2}$ . That is
${V}_{1}·{V}_{2}={v}_{1}{v}_{2}cos\theta ={a}_{1}{a}_{2}+{b}_{1}{b}_{2}+{c}_{1}{c}_{2}={V}_{2}·{V}_{1}$
Note the relations:
$\begin{array}{cc}\hfill \left({V}_{1}+{V}_{2}\right)·{V}_{3}& ={V}_{1}·{V}_{3}+{V}_{2}·{V}_{3}\hfill \\ \multicolumn{1}{c}{{V}_{1}·\left({V}_{2}+{V}_{3}\right)}& ={V}_{1}·{V}_{2}+{V}_{1}·{V}_{3}\hfill \\ \multicolumn{1}{c}{}\end{array}$
If ${V}_{1}$ is perpendicular to ${V}_{2}$ then ${V}_{1}·{V}_{2}=0$ , and if ${V}_{1}$ is parallel to ${V}_{2}$ then ${V}_{1}·{V}_{2}=|{V}_{1}|\phantom{\rule{0.2em}{0ex}}|{V}_{2}|$ . In particular:
$\begin{array}{cc}\hfill i·i& =j·j=k·k=1\hfill \\ \multicolumn{1}{c}{i·j}& =j·k=k·i=0\hfill \\ \multicolumn{1}{c}{}\end{array}$

### 2.2.4 The Vector or Cross Product of Two Vectors

This product is represented as ${V}_{1}×{V}_{2}$ and is defined as
${V}_{1}×{V}_{2}=|{V}_{1}||{V}_{2}|\phantom{\rule{0.2em}{0ex}}sin\theta \phantom{\rule{0.2em}{0ex}}1$
where $\theta$ is the angle from ${V}_{1}$ to ${V}_{2}$ and $1$ is a unit vector perpendicular to the plane of ${V}_{1}$ and ${V}_{2}$ and so directed that a right-handed screw driven in the direction of $1$ would carry ${V}_{1}$ into ${V}_{2}$ . Note that
$tan\theta =\frac{|{V}_{1}×{V}_{2}|}{{V}_{1}·{V}_{2}}$
The following rules apply to vector products:
$\begin{array}{cc}\hfill {V}_{1}×{V}_{2}& =-{V}_{2}×{V}_{1}\hfill \\ \multicolumn{1}{c}{{V}_{1}×\left({V}_{2}+{V}_{3}\right)}& ={V}_{1}×{V}_{2}+{V}_{1}×{V}_{3}\hfill \\ \multicolumn{1}{c}{\left({V}_{1}+{V}_{2}\right)×{V}_{3}}& ={V}_{1}×{V}_{3}+{V}_{2}×{V}_{3}\hfill \\ \multicolumn{1}{c}{{V}_{1}×\left({V}_{2}×{V}_{3}\right)}& ={V}_{2}\left({V}_{3}·{V}_{1}\right)-{V}_{3}\left({V}_{1}·{V}_{2}\right)\hfill \\ \multicolumn{1}{c}{i×i=0,\mathrm{ }j×j}& =0,\mathrm{ }k×k=0\hfill \\ \multicolumn{1}{c}{i×j=k,\mathrm{ }j×k}& =i,\mathrm{ }k×i=j\hfill \end{array}$
If ${V}_{1}={a}_{1}i+{b}_{1}j+{c}_{1}k$ , ${V}_{2}={a}_{2}i+{b}_{2}j+{c}_{2}k$ , and ${V}_{3}={a}_{3}i+{b}_{3}j+{c}_{3}k$ , then
${V}_{1}×{V}_{2}=|\begin{array}{lll}i\hfill & j\hfill & k\hfill \\ {a}_{1}\hfill & {b}_{1}\hfill & {c}_{1}\hfill \\ {a}_{2}\hfill & {b}_{2}\hfill & {c}_{2}\hfill \end{array}|=\left({b}_{1}{c}_{2}-{b}_{2}{c}_{1}\right)\mathrm{i}+\left({c}_{1}{a}_{2}-{c}_{2}{a}_{1}\right)\mathrm{j}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}\right)\mathrm{k}$
Note that, since ${V}_{1}×{V}_{2}=-{V}_{2}×{V}_{1}$ , the vector product is not commutative.

### 2.2.5 Scalar Triple Product

There is only one possible interpretation of the expression ${V}_{1}·{V}_{2}×{V}_{3}$ and that is ${V}_{1}·\left({V}_{2}×{V}_{3}\right)$ which is a scalar. This product is called the scalar triple product and is written as $\left[{V}_{1}{V}_{2}{V}_{3}\right]$ . Further
$\begin{array}{lll}\left[{V}_{1}{V}_{2}{V}_{3}\right]\hfill & =\hfill & {V}_{1}\cdot \left({V}_{2}×{V}_{3}\right)=\left({V}_{1}×{V}_{2}\right)\cdot {V}_{3}={V}_{2}\cdot \left({V}_{3}×{V}_{1}\right)\hfill \\ \hfill & =\hfill & |\begin{array}{lll}{a}_{1}\hfill & {b}_{1}\hfill & {c}_{1}\hfill \\ {a}_{2}\hfill & {b}_{2}\hfill & {c}_{2}\hfill \\ {a}_{3}\hfill & {b}_{3}\hfill & {c}_{3}\hfill \end{array}|\hfill \\ \hfill & =\hfill & |{V}_{1}||{V}_{2}||{V}_{3}|cos\varphi sin\theta ,\hfill \end{array}$
where $\theta$ is the angle between ${V}_{2}$ and ${V}_{3}$ and $\phi$ is the angle between ${V}_{1}$ and the normal to the plane of ${V}_{2}$ and ${V}_{3}$ . The determinant indicates that it can be considered as the volume of the parallelepiped whose three determining edges are ${V}_{1}$ , ${V}_{2}$ , and ${V}_{3}$ . Note that cyclic permutation of the subscripts does not change the value of the scalar triple product: $\left[{V}_{1}{V}_{2}{V}_{3}\right]=\left[{V}_{2}{V}_{3}{V}_{1}\right]=\left[{V}_{3}{V}_{1}{V}_{2}\right]$ but $\left[{V}_{1}{V}_{2}{V}_{3}\right]=-\left[{V}_{2}{V}_{1}{V}_{3}\right]$ and $\left[{V}_{1}{V}_{1}{V}_{2}\right]\equiv 0$ .

### 2.2.6 Vector Triple Product

The product ${V}_{1}×\left({V}_{2}×{V}_{3}\right)$ defines the vector triple product. The parentheses are vital to the definition.
$\begin{array}{lll}{V}_{1}×\left({V}_{2}×{V}_{3}\right)\hfill & =\hfill & \left({V}_{1}\cdot {V}_{3}\right){V}_{2}-\left({V}_{1}\cdot {V}_{2}\right){V}_{3}\hfill \\ \hfill & =\hfill & |\begin{array}{lll}i\hfill & j\hfill & k\hfill \\ {a}_{1}\hfill & {b}_{1}\hfill & {c}_{1}\hfill \\ |\begin{array}{ll}{b}_{2}\hfill & {c}_{2}\hfill \\ {b}_{3}\hfill & {c}_{3}\hfill \end{array}|\hfill & |\begin{array}{ll}{c}_{2}\hfill & {a}_{2}\hfill \\ {c}_{3}\hfill & {a}_{3}\hfill \end{array}|\hfill & |\begin{array}{ll}{a}_{2}\hfill & {b}_{2}\hfill \\ {a}_{3}\hfill & {b}_{3}\hfill \end{array}|\hfill \end{array}|\hfill \end{array}$
This is a vector, perpendicular to ${V}_{1}$ , lying in the plane of ${V}_{2}$ and ${V}_{3}$ . Similarly
$\begin{array}{lll}\left({V}_{1}×{V}_{2}\right)×{V}_{3}\hfill & =\hfill & |\begin{array}{lll}i\hfill & j\hfill & k\hfill \\ |\begin{array}{ll}{b}_{1}\hfill & {c}_{1}\hfill \\ {b}_{2}\hfill & {c}_{2}\hfill \end{array}|\hfill & |\begin{array}{ll}{c}_{1}\hfill & {a}_{1}\hfill \\ {c}_{2}\hfill & {a}_{2}\hfill \end{array}|\hfill & |\begin{array}{ll}{a}_{1}\hfill & {b}_{1}\hfill \\ {a}_{2}\hfill & {b}_{2}\hfill \end{array}|\hfill \\ {a}_{3}\hfill & {b}_{3}\hfill & {c}_{3}\hfill \end{array}|\hfill \\ {V}_{1}×\left({V}_{2}×{V}_{3}\right)\hfill & +\hfill & {V}_{2}×\left({V}_{3}×{V}_{1}\right)+{V}_{3}×\left({V}_{1}×{V}_{2}\right)\equiv 0\hfill \end{array}$
If ${V}_{1}×\left({V}_{2}×{V}_{3}\right)=\left({V}_{1}×{V}_{2}\right)×{V}_{3}$ then ${V}_{1},{V}_{2},{V}_{3}$ form an <i>orthogonal set</i>. Thus $\left\{i,j,k\right\}$ form an orthogonal set.

Page 1 of 1
1/1

Entry Display
This is where the entry will be displayed

#### Other ChemNetBase Products

 You are not within the network of a subscribing institution.Please sign in with an Individual User account to continue.Note that Workspace accounts are not valid.

Confirm Log Out
Are you sure?
Your personal workspace allows you to save and access your searches and bookmarks.

Are you sure?

 You have entered your Individual User account sign in credentials instead of Workspace credentials. While using this network, a personal workspace account can be created to save your bookmarks and search preferences for later use. Click the help icon for more information on the differences between Individual User accounts and Workspace accounts.
My Account

 Username Title [Select]DrProfMissMrsMsMrMx [Select]DrProfMissMrsMsMrMx First Name (Given) Last Name (Family) Email address

Searching for Chemicals and Properties

The CRC Handbook of Chemistry and Physics (HBCP) contains over 700 tables in over 450 documents which may be divided into several pages, all categorised into 17 major subject areas. The search on this page works by searching the content of each page individually, much like any web search. This provides a challenge if you want to search for multiple terms and those terms exist on different pages, or if you use a synonym/abbreviation that does not exist in the document.

We use metadata to avoid some of these issues by including certain keywords invisibly behind each table. Whilst this approach works well in many situations, like any web search it relies in the terms you have entered existing in the document with the same spelling, abbreviation etc.

Since chemical compounds and their properties are immutable, a single centralised database has been created from all chemical compounds throughout HBCP. This database contains every chemical compound and over 20 of the most common physical properties collated from each of the >700 tables. What's more, the properties can be searched numerically, including range searching, and you can even search by drawing a chemical structure. A complete list of every document table in which the compound occurs is listed, and are hyperlinked to the relevant document table.

The 'Search Chemicals' page can be found by clicking the flask icon in the navigation bar at the top of this page. For more detailed information on how to use the chemical search, including adding properties, saving searches, exporting search results and more, click the help icon in to top right of this page, next to the welcome login message.

Below is an example of a chemical entry, showing its structure, physical properties and document tables in which it appears.

We have developed this cookie policy (the “Cookie Policy”) in order to explain how we use cookies and similar technologies (together, “Cookies”) on this website (the “Website”) and to demonstrate our firm commitment to the privacy of your personal information.

The first time that you visit our Website, we notify you about our use of Cookies through a notification banner. By continuing to use the Website, you consent to our use of Cookies as described in this Cookie Policy. However, you can choose whether or not to continue accepting Cookies at any later time. Information on how to manage Cookies is set out later in this Cookie Policy.

Cookies are small text files containing user IDs that are automatically placed on your computer or other device by when you visit a website. The Cookies are stored by the internet browser. The browser sends the Cookies back to the website on each subsequent visit, allowing the website to recognise your computer or device. This recognition enables the website provider to observe your activity on the website, deliver a personalised, responsive service and improve the website.

## Cookies We Use and Their Purpose

‘Strictly Necessary’ Cookies enable you to move around the Website and use essential features. For example, if you log into the Website, we use a Cookie to keep you logged in and allow you to access restricted areas, without you having to repeatedly enter your login details. If you are registering for or purchasing a product or service, we will use Cookies to remember your information and selections, as you move through the registration or purchase process.

Strictly Necessary Cookies are necessary for our Website to provide you with a full service. If you disable them, certain essential features of the Website will not be available to you and the performance of the Website will be impeded.

‘Performance’ Cookies collect information about how you use our Website, for example which pages you visit and if you experience any errors. These Cookies don’t collect any information that could identify you – all the information collected is anonymous. We may use these Cookies to help us understand how you use the Website and assess how well the Website performs and how it could be improved.

‘Functionality’ Cookies enable a website to provide you with specific services or a customised experience. We may use these Cookies to provide you with services such as watching a video or adding user comments. We may also use such Cookies to remember changes you make to your settings or preferences (for example, changes to text size or your choice of language or region) or offer you time-saving or personalised features.

You can control whether or not Functionality Cookies are used, but disabling them may mean we are unable to provide you with some services or features of the Website.

## First and Third Party Cookies

The Cookies placed on your computer or device include ‘First Party’ Cookies, meaning Cookies that are placed there by us, or by third party service providers acting on our behalf. Where such Cookies are being managed by third parties, we only allow the third parties to use the Cookies for our purposes, as described in this Cookie Policy, and not for their own purposes.

You always have a choice over whether or not to accept Cookies. When you first visit the Website and we notify you about our use of Cookies, you can choose not to consent to such use. If you continue to use the Website, you are consenting to our use of Cookies for the time being. However, you can choose not to continue accepting Cookies at any later time. In this section, we describe ways to manage Cookies, including how to disable them.

You can manage Cookies through the settings of your internet browser. You can choose to block or restrict Cookies from being placed on your computer or device. You can also review periodically review the Cookies that have been placed there and disable some or all of them.

Please be aware that if you choose not to accept certain Cookies, it may mean we are unable to provide you with some services or features of the Website.

In order to keep up with changing legislation and best practice, we may revise this Cookie Policy at any time without notice by posting a revised version on this Website. Please check back periodically so that you are aware of any changes.

## Questions or Concerns

You can also contact the Privacy Officer for the Informa PLC group at [email protected].

Here is a list of cookies we have defined as 'Strictly Necessary':

### Taylor and Francis 'First Party' Cookies

JSESSIONID

Here is a list of the cookies we have defined as 'Performance'.

_ga

_gid

_gat

Accessibility

The Voluntary Product Accessibility Template (VPAT) is a self-assessment document which discloses how accessible Information and Communication Technology products are in accordance with global standards.

The VPAT disclosure templates do not guarantee product accessibility but provide transparency around the product(s) and enables direction when accessing accessibility requirements.

Taylor & Francis has chosen to complete the International version of VPAT which encompasses Section 508 (US), EN 301 549 (EU) and WCAG2.1 (Web Content Accessibility Guidelines) for its products.