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(1) Definition :
The vector product or cross product of two vectors is defined as a vector
having a magnitude equal to the product of the magnitudes of two vectors with
the sine of angle between them, and direction perpendicular to the plane
containing the two vectors in accordance with right hand screw rule.
\[\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}\]
Thus,
if \[\overrightarrow{A}\] and \[\overrightarrow{B}\] are two vectors, then
their vector product written as \[\overrightarrow{A}\times \overrightarrow{B}\]
is a vector \[\overrightarrow{C}\] defined by
\[\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta
\,\hat{n}\]
(1) Definition : The scalar product (or dot product) of two vectors is defined as the product of the magnitude of two vectors with cosine of angle between them.
Thus if there are two vectors \[\overrightarrow{A}\]and \[\overrightarrow{B}\] having angle \[\theta \] between them, then their scalar product written as \[\overrightarrow{A}\,.\,\overrightarrow{B}\] is defined as \[\overrightarrow{A}\,.\,\overrightarrow{B}\] \[=AB\,\cos \theta \]
 
(2) Properties : (i) It is always a scalar which is positive if angle between the vectors is acute (i.e., < 90°) and negative if angle between them is obtuse (i.e. 90°<q < 180°).
(ii) It is commutative, i.e. \[\overrightarrow{A}\,.\,\overrightarrow{B}\,=\,\overrightarrow{B}\,.\,\overrightarrow{A}\]
(iii) It is distributive, i.e. \[\overrightarrow{A}\,.\,(\overrightarrow{B}+\overrightarrow{C})\,=\overrightarrow{A}\,.\,\overrightarrow{B}\,+\overrightarrow{A}\,.\,\overrightarrow{C}\]
(iv) As by definition \[\overrightarrow{A}\,.\,\overrightarrow{B}=AB\,\cos \theta \]
The angle between the vectors \[\theta ={{\cos }^{-1}}\left[ \frac{\overrightarrow{A}\,.\,\overrightarrow{B}}{AB} \right]\]
(v) Scalar product of two vectors will be maximum when \[\cos \theta =\max =1,\] i.e. \[\theta ={{0}^{o}},\] i.e., vectors are parallel
\[{{(\overrightarrow{A}\,.\,\overrightarrow{B})}_{\max }}=AB\]
(vi) Scalar product of two vectors will be minimum when \[|\cos \theta |\,=\min =0,\,\]i.e. \[\theta ={{90}^{o}}\]
\[{{(\overrightarrow{A}\,.\,\overrightarrow{B})}_{\min }}=0\]
i.e. if the scalar product of two nonzero vectors vanishes the vectors are orthogonal.
(vii) The scalar product of a vector by itself is termed as self dot product and is given by \[{{(\overrightarrow{A})}^{2}}=\overrightarrow{A}\,.\,\overrightarrow{A}=AA\,\cos \theta ={{A}^{2}}\]
i.e. \[A=\sqrt{\overrightarrow{A}\,.\,\overrightarrow{A}}\]
(viii) In case of unit vector \[\hat{n}\]
\[\hat{n}\,.\,\hat{n}=1\times 1\times \cos 0=1\] so \[\hat{n}\,.\,\hat{n}\,=\hat{i}\,.\,\hat{i}\,=\hat{j}\,.\,\hat{j}\,=\hat{k}\,.\,\hat{k}\,=1\]
(ix) In case of orthogonal unit vectors \[\hat{i},\,\hat{j}\] and \[\hat{k},\] \[\hat{i}\,.\,\hat{j}\,=\hat{j}\,.\,\hat{k}\,=\hat{k}\,.\,\hat{i}\,=1\times 1\cos 90{}^\circ =0\]
(x) In terms of components
\[\overrightarrow{A}\,.\,\overrightarrow{B}\,=\,(\overrightarrow{i}{{A}_{x}}+\overrightarrow{j}{{A}_{y}}+\overrightarrow{k}{{A}_{z}})\,.\,(\overrightarrow{i}{{B}_{x}}+\overrightarrow{j}{{B}_{y}}+\overrightarrow{k}{{B}_{z}})\]\[=[{{A}_{x}}{{B}_{x}}+{{A}_{y}}{{B}_{y}}+{{A}_{Z}}{{B}_{z}}]\]
(3) Example : (i) Work W : In physics for constant force work is defined as, \[W=Fs\cos \theta \] …(i)
But by definition of scalar product of two vectors, \[\overrightarrow{F}.\,\overrightarrow{s}=Fs\cos \theta \] …(ii)
So from eqn (i) and (ii) \[W=\overrightarrow{F}.\overrightarrow{s}\] i.e. work is the scalar product of force with displacement.
(ii) Power P :
As \[W=\overrightarrow{F}\,.\,\overrightarrow{s}\] or \[\frac{dW}{dt}=\overrightarrow{F}\,.\,\frac{d\overrightarrow{s}}{dt}\] [As \[\overrightarrow{F}\] is constant]
or \[P=\overrightarrow{F}\,.\,\overrightarrow{v}\] i.e., power is the scalar product of force with velocity. \[\left[ \text{As}\frac{dW}{dt}=P\,\text{and}\,\frac{d\overrightarrow{s}}{dt}=\overrightarrow{v} \right]\]
(iii) Magnetic Flux \[\varphi \]:
Magnetic flux through an area is given by \[d\varphi =B\,ds\cos \theta \] …(i)
But by definition of scalar product \[\overrightarrow{B}\,.\,d\,\overrightarrow{s}=Bds\,\cos \theta \] ...(ii)
So from eqn (i) and (ii) we have
\[d\varphi =\overrightarrow{B}\,.\,d\,\overrightarrow{s}\] or \[\varphi =\int_{{}}^{{}}{\overrightarrow{B}\,.\,d\overrightarrow{s}}\]
(iv) Potential energy of a dipole U : If an electric dipole of moment \[\overrightarrow{p}\] is situated in an electric field \[\overrightarrow{E}\] or a magnetic dipole of moment \[\overrightarrow{M}\] in a field of induction \[\overrightarrow{B},\] the potential energy of the dipole is given by :
\[{{U}_{E}}=-\overrightarrow{p}\,.\,\overrightarrow{E}\] and \[{{U}_{B}}=-\overrightarrow{M}\,.\,\overrightarrow{B}\]
\[\overrightarrow{R}={{\overrightarrow{R}}_{x}}+{{\overrightarrow{R}}_{y}}+{{\overrightarrow{R}}_{z}}q\] or \[\overrightarrow{R}={{R}_{x}}\hat{i}+{{R}_{y}}\hat{j}+{{R}_{z}}\hat{k}\]
If \[\overrightarrow{R}\] makes an angle a with x axis, b with y axis and \[\gamma \]with z axis, then
\[\Rightarrow q\] \[\cos \alpha =\frac{{{R}_{x}}}{R}=\frac{{{R}_{x}}}{\sqrt{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}}=l\]
\[\Rightarrow \] \[\cos \beta =\frac{{{R}_{y}}}{R}=\frac{{{R}_{y}}}{\sqrt{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}}=m\]
\[\Rightarrow \] \[\cos \gamma =\frac{{{R}_{z}}}{R}=\frac{{{R}_{z}}}{\sqrt{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}}=n\]
Where l, m, n are called Direction Cosines of the vector \[\overrightarrow{R}\] and \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=\]\[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =\frac{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}{R_{x}^{2}+R_{y}^{2}+R_{z}^{2}}=1\]
Note :
The phenomenon of resonance was put forward by Heisenberg to explain the properties of certain molecules.
In case of certain molecules, a single Lewis structure cannot explain all the properties of the molecule. The molecule is then supposed to have many structures, each of which can explain most of the properties of the molecule but none can explain all the properties of the molecule. The actual structure is in between of all these contributing structures and is called resonance hybrid and the different individual structures are called resonating structures or canonical forms. This phenomenon is called resonance.
To illustrate this, consider a molecule of ozone \[{{O}_{3}}\]. Its structure can be written as
  \[\underset{(a)}{\mathop{\begin{array}{*{35}{l}} \,\,\,\,\,\,O \\ {} \\ O\,\,\,\,\,\,\,\,O \\ \end{array}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underset{(b)}{\mathop{\begin{array}{*{35}{l}} \,\,\,\,\,\,O \\ {} \\ O\,\,\,\,\,\,\,\,O \\ \end{array}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underset{(c)}{\mathop{\begin{array}{*{35}{l}} \,\,\,\,\,\,O \\ {} \\ O\,\,\,\,\,\,\,\,O \\ \end{array}}}\,\] As a resonance hybrid of above two structures (a) and (b). For simplicity, ozone may be represented by structure (c), which shows the resonance hybrid having equal bonds between single and double.
Resonance is shown by benzene, toluene, O3, allenes (>C = C = C<), CO, CO2, \[CO_{3}^{-}\], SO3, NO, NO2 while it is not shown by H2O2, H2O, NH3, CH4, SiO2.
As a result of resonance, the bond lengths of single and double bond in a molecule become equal e.g. O?O bond lengths in ozone or C?O bond lengths in \[CO_{3}^{2}\]ion.
The resonance hybrid has lower energy and hence greater stability than any of the contributing structures.
Greater is the number of canonical forms especially with nearly same energy, greater is the stability of the molecule.
Difference between the energy of resonance hybrid and that of the most stable of the resonating structures (having least energy) is called resonance energy. Thus,
Resonance energy = Energy of resonance hybrid ? Energy of the most stable of resonating structure.
In the case of molecules or ions having resonance, the bond order changes and is calculated as follows, \[\text{Bond order }=\frac{\text{Total no}\text{. of bonds between two atoms in all the structures}}{\text{Total no}\text{. of resonating structures}}\]   In benzene \[\begin{matrix} \text{ } \\ \text{ } \\ \text{ } \\ \end{matrix}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\overset{\,\,\,\,}{\longleftrightarrow}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\] \[\text{Bond order }=\frac{\text{double bond }+s\text{ingle bond}}{2}=\frac{2+1}{2}=1.5\] In carbonate ion \[\begin{array}{*{35}{l}} \,\,\,\,\,\,{{O}^{-}} \\ \,\,\,\,\,\,\,| \\ \,\,\,\,\,\,C \\ \,\,\,//\,\,\,\,\backslash \\ O\,\,\,\,\,\,\,\,{{O}^{-}} \\ \end{array}\,\,\,\,\,\,\overset{\,\,}{\longleftrightarrow}\,\,\,\,\,\begin{array}{*{35}{l}} \,\,\,\,\,\,\,\,O \\ \,\,\,\,\,\,\,\,\,|\,|\, \\ \,\,\,\,\,\,\,\,C \\ \,\,\,\,\,\,/\,\,\,\,\backslash \\ ^{-}O\,\,\,\,\,\,{{O}^{-}} \\ \end{array}\,\,\,\,\overset{\,\,}{\longleftrightarrow}\,\,\,\,\begin{array}{*{35}{l}} \,\,\,\,\,\,\,\,{{O}^{-}} \\ \,\,\,\,\,\,\,\,\,| \\ \,\,\,\,\,\,\,\,C \\ \,\,\,\,\,\,/\,\,\,\,\backslash \,\backslash \\ ^{-}O\,\,\,\,\,\,\,O \\ \end{array}\] \[\text{Bond order }=\frac{2+1+1}{3}=1.33\]
Consider a vector \[\overrightarrow{R}\,\] in X-Y plane as shown in fig. If we draw orthogonal vectors \[{{\overrightarrow{R}}_{x}}\] and \[{{\overrightarrow{R}}_{y}}\] along x and \[y\]axes respectively, by law of vector addition, \[\vec{R}={{\vec{R}}_{x}}+{{\vec{R}}_{y}}\]
Now as for any vector \[\overrightarrow{A}=A\,\hat{n}\]
so, \[{{\overrightarrow{R}}_{x}}=\hat{i}{{R}_{x}}\] and \[{{\overrightarrow{R}}_{y}}=\hat{j}{{R}_{y}}\]
so \[\overrightarrow{R}=\hat{i}{{R}_{x}}+\hat{j}{{R}_{y}}\] ...(i)
But from figure \[{{R}_{x}}=R\cos \theta \] ...(ii)
and \[{{R}_{y}}=R\sin \theta \] ...(iii)
Since R and q are usually known, Equation (ii) and (iii) give the magnitude of the components of \[\overrightarrow{R}\] along x and y-axes respectively.
Here it is worthy to note once a vector is resolved into its components, the components themselves can be used to specify the vector as
(1) The magnitude of the vector\[\overrightarrow{R}\] is obtained by squaring and adding equation (ii) and (iii), i.e. \[R=\sqrt{R_{x}^{2}+R_{y}^{2}}\]
(2) The direction of the vector \[\overrightarrow{R}\] is obtained by dividing equation (iii) by (ii), i.e.
\[\tan \theta =({{R}_{y}}/{{R}_{x}})\] or \[\theta ={{\tan }^{-1}}({{R}_{y}}/{{R}_{x}})\]
Since, \[\overrightarrow{A}-\overrightarrow{B}=\overrightarrow{A}+(-\overrightarrow{B})\] and
\[|\overrightarrow{A}+\overrightarrow{B}|\,=\, \sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\] \[\Rightarrow \]
\[|\overrightarrow{A}-\overrightarrow{B}|\,=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \,({{180}^{o}}-\theta )}\]
Since, \[\cos \,(180-\theta )=-\cos \theta \]
\[\Rightarrow \] \[|\overrightarrow{A}-\overrightarrow{B}|\,=\,\sqrt{{{A}^{2}}+{{B}^{2}}-2AB\cos \theta }\]
\[\tan {{\alpha }_{1}}=\frac{B\sin \theta }{A+B\cos \theta }\]
and \[\tan {{\alpha }_{2}}=\frac{B\sin \,(180-\theta )}{A+B\cos \,(180-\theta )}\]
But \[\sin (180-\theta )=\sin \theta \] and \[\cos (180-\theta )=-\cos \theta \] \[\Rightarrow \]
\[\tan {{\alpha }_{2}}=\frac{B\sin \theta }{A-B\cos \theta }\]
The concept of hybridization was introduced by Pauling and Slater. HybridizationIt is defined as the intermixing of dissimilar orbitals of the same atom but having slightly different energies to form same number of new orbitals of equal energies and identical shapes. The new orbitals so formed are known as hybrid orbitals.
Characteristics of hybridization
(1) Only orbitals of almost similar energies and belonging to the same atom or ion undergoes hybridization.
(2) Hybridization takes place only in orbitals, electrons are not involved in it.
(3) The number of hybrid orbitals produced is equal to the number of pure orbitals, mixed during hybridization.
(4) In the excited state, the number of unpaired electrons must correspond to the oxidation state of the central atom in the molecule.
(5) Both half filled orbitals or fully filled orbitals of equivalent energy can involve in hybridization.
(6) Hybrid orbitals form only sigma bonds.
(7) Orbitals involved in p bond formation do not participate in hybridization.
(8) Hybridization never takes place in an isolated atom but it occurs only at the time of bond formation.
(9) The hybrid orbitals are distributed in space as apart as possible resulting in a definite geometry of molecule.
(10) Hybridized orbitals provide efficient overlapping than overlapping by pure s, p and d-orbitals.
(11) Hybridized orbitals possess lower energy.
How to determine type of hybridization : The structure of any molecule can be predicted on the basis of hybridization which in turn can be known by the following general formulation,
\[H=\frac{1}{2}(V+M-C+A)\]
Where H = Number of orbitals involved in hybridization viz. 2, 3, 4, 5, 6 and 7, hence nature of hybridization will be \[sp,\text{ }s{{p}^{2}},\text{ }s{{p}^{3}},\text{ }s{{p}^{3}}d,\text{ }s{{p}^{3}}{{d}^{2}},\text{ }s{{p}^{3}}{{d}^{3}}\] respectively.
V = Number electrons in valence shell of the central atom,
M = Number of monovalent atom
C = Charge on cation,
A = Charge on anion
It was developed by Heitler and London in 1927 and modified by Pauling and Slater in 1931.
(1) To form a covalent bond, two atoms must come close to each other so that orbitals of one overlaps with the other.
(2) Orbitals having unpaired electrons of anti spin overlaps with each other.
(3) After overlapping a new localized bond orbital is formed which has maximum probability of finding electrons.
(4) Covalent bond is formed due to electrostatic attraction between radii and the accumulated electrons cloud and by attraction between spins of anti spin electrons.
(5) Greater is the overlapping, lesser will be the bond length, more will be attraction and more will be bond energy and the stability of bond will also be high.
(6) The extent of overlapping depends upon: Nature of orbitals involved in overlapping, and nature of overlapping.
(7) More closer the valence shells are to the nucleus, more will be the overlapping and the bond energy will also be high.
(8) Between two sub shells of same energy level, the sub shell more directionally concentrated shows more overlapping. Bond energy : \[2s-2s\]< \[2s-2p\]< \[2p-2p\]
(9) \[s\]-orbitals are spherically symmetrical and thus show only head on overlapping. On the other hand, \[p\]-orbitals are directionally concentrated and thus show either head on overlapping or lateral overlapping. Overlapping of different type gives sigma (s) and pi (p) bond.
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pointing towards the negative end. Dipole moment has both magnitude and direction and therefore it is a vector quantity.
Symmetrical polyatomic molecules are not polar so they do not have any value of dipole moment. 
Unsymmetrical polyatomic molecules always have net value of dipole moment, thus such molecules are polar in nature. H2O, CH3Cl, NH3, etc are polar molecules as they have some positive values of dipole moments.
\[\mu =0\]due to unsymmetry
(1) Dipole moment is an important factor in determining the geometry of molecules. Molecular geometry and dipole moment
