If we say that P is the point \[\mathbf{r}\], then we mean that the position vector of \[P\] is \[\mathbf{r}\] with respect to some origin \[O\].
(1) \[\overrightarrow{AB}\] in terms of the position vectors of points A and B : If \[\mathbf{a}\] and b are position vectors of points A and B respectively. Then, \[\overrightarrow{OA}=\mathbf{a},\,\overrightarrow{OB}=\mathbf{b}\]
\[\therefore \] \[\overrightarrow{AB}\] = (Position vector of B) – (Position vector of A)
\[=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}\]
(2) Position vector of a dividing point : The position vectors of the points dividing the line \[AB\]in the ratio m : n internally or externally are \[\frac{m\mathbf{b}+n\mathbf{a}}{m+n}\] or \[\frac{m\mathbf{b}-n\mathbf{a}}{m-n}\].
(ii) Parallelogram law of addition : If in a parallelogram \[OACB,\] \[\overrightarrow{OA}=\mathbf{a},\,\overrightarrow{OB}=\mathbf{b}\] and \[\overrightarrow{OC}\,=\mathbf{c}\]
Then \[\overrightarrow{OA}+\overrightarrow{OB}=\overrightarrow{OC}\] i.e., \[\mathbf{a}+\mathbf{b}=\mathbf{c}\ \], where OC is a diagonal of the parallelogram OABC.
(iii) Addition in component form : If the vectors are defined in terms of \[\mathbf{i,}\,\,\,\mathbf{j}\] and \[\mathbf{k,}\] i.e., if \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\] and \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\], then their sum is defined as \[\mathbf{a}+\mathbf{b}=({{a}_{1}}+{{b}_{1}})\mathbf{i}+({{a}_{2}}+{{b}_{2}})\mathbf{j}+({{a}_{3}}+{{b}_{3}})\mathbf{k}\].
Properties of vector addition : Vector addition has the following properties.
(a) Binary operation : The sum of two vectors is always a vector.
(b) Commutativity : For any two vectors \[\mathbf{a}\] and \[\mathbf{b}\], \[\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}\].
(c) Associativity : For any three vectors \[\mathbf{a},\,\mathbf{b}\] and \[\mathbf{c}\], \[\mathbf{a}+(\mathbf{b}+\mathbf{c})=(\mathbf{a}+\mathbf{b})+\mathbf{c}\].
(d) Identity : Zero vector is the identity for addition. For any vector \[\mathbf{a},\,\,\mathbf{0}+\mathbf{a}=\mathbf{a}=\mathbf{a}+\mathbf{0}\]
(e) Additive inverse : For every vector \[\mathbf{a}\] its negative vector \[-\mathbf{a}\] exists such that \[\mathbf{a}+(-\mathbf{a})=(-\mathbf{a})+\mathbf{a}=\mathbf{0}\] i.e., \[(-\mathbf{a})\] is the additive inverse of the vector \[\mathbf{a}\].
(2) Subtraction of vectors : If \[\mathbf{a}\]and \[\mathbf{b}\] are two vectors, then their subtraction \[\mathbf{a}-\mathbf{b}\] is defined as\[\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})\] where \[-\mathbf{b}\] is the negative of \[\mathbf{b}\] having magnitude equal to that of \[\mathbf{b}\] and direction opposite to \[\mathbf{b}\]. If \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\], \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\]
Then \[\mathbf{a}-\mathbf{b}=({{a}_{1}}-{{b}_{1}})\mathbf{i}+({{a}_{2}}-{{b}_{2}})\mathbf{j}+({{a}_{3}}-{{b}_{3}})\mathbf{k}\].
Properties of vector subtraction
(i) \[\mathbf{a}-\mathbf{b}\ne \mathbf{b}-\mathbf{a}\]
(ii) \[(\mathbf{a}-\mathbf{b})-\mathbf{c}\ne \mathbf{a}-(\mathbf{b}-\mathbf{c})\]
(iii) Since any one side of a triangle is less than the sum and greater than the difference of the other two sides, so for any two vectors \[a\] and \[b,\] we have
(a) \[|\mathbf{a}+\mathbf{b}|\,\le \,|\mathbf{a}|+|\mathbf{b}|\]
(b) \[|\mathbf{a}+\mathbf{b}|\,\ge \,|\mathbf{a}|-|\mathbf{b}|\]
(c) \[|\mathbf{a}-\mathbf{b}|\,\le \,|\mathbf{a}|+|\mathbf{b}|\]
(d) \[|\mathbf{a}-\mathbf{b}|\,\ge \,|\mathbf{a}|\,-\,|\mathbf{b}|\]
(3) Multiplication of a vector by a scalar : If \[\mathbf{a}\] is a vector and \[m\] is a scalar (i.e., a real number) then \[m\mathbf{a}\] is a vector whose magnitude is \[m\] times that of \[\mathbf{a}\] and whose direction is the same as that of \[\mathbf{a}\], if m is positive and opposite to that of \[\mathbf{a}\], if m is negative.
Properties of Multiplication of vectors by a scalar : The following are properties of multiplication of vectors by scalars, for vectors \[\mathbf{a},\,\mathbf{b}\] and scalars \[m,\,\,n\].
(i) \[m(-\mathbf{a})=(-m)\,\mathbf{a}=-(m\mathbf{a})\]
(ii) \[(-m)\,(-\mathbf{a})=m\mathbf{a}\]
(iii) \[m\,(n\mathbf{a})=(mn)\,\mathbf{a}=n(m\mathbf{a})\]
(iv) \[(m+n)\,\mathbf{a}=m\mathbf{a}+n\mathbf{a}\]
(v) \[m\,(\mathbf{a}+\mathbf{b})=m\mathbf{a}+m\mathbf{b}\]
(4) Resultant of two forces
Let \[\overrightarrow{P}\] and \[\overrightarrow{Q}\] be two forces and \[\overrightarrow{R}\] be the resultant of these two forces then, \[\overrightarrow{R}\,=\overrightarrow{P}+\overrightarrow{Q}\]
\[|\overrightarrow{R}|=R=\sqrt{{{P}^{2}}+{{Q}^{2}}+2PQ\,\cos \theta }\]
where \[|\overrightarrow{P}|=P,\,|\overrightarrow{Q}|=Q,\]
Also, \[\tan \alpha =\frac{Q\sin \theta }{P+Q\cos \theta }\]
Deduction : When \[|\overrightarrow{P}|=|\overrightarrow{Q}|\], more... 
Magnitude or modulus of \[\mathbf{a}\] is expressed as
\[|\mathbf{a}|=|\overrightarrow{AB}|=AB\]. 
\[SP=ePM\]\[\Rightarrow \] \[S{{P}^{2}}={{e}^{2}}P{{M}^{2}}\]
\[\Rightarrow \] \[{{(h-\alpha )}^{2}}+{{(k-\beta )}^{2}}={{e}^{2}}{{\left( \frac{Ah+Bk+C}{\sqrt{{{A}^{2}}+{{B}^{2}}}} \right)}^{2}}\]
Thus the locus of \[(h,k)\] is \[{{(x-\alpha )}^{2}}+{{(y-\beta )}^{2}}=\]\[{{e}^{2}}\frac{{{(Ax+By+C)}^{2}}}{({{A}^{2}}+{{}^{2}})}\]
which is general equation of second degree. 
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