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Friday, 8 August 2014

CHAPTER 17-Properties of Dot Product

From the definition of the dot product, we can make certain useful observations about its properties.
(i) The angle \theta  between two vectors \vec a and \vec b is given by
\cos \theta  = \dfrac{{\vec a \cdot \vec b}}{{\left| {\vec a} \right|\;\left| {\vec b} \right|}}
(ii) \left| {\vec a \cdot \vec b} \right| \le \left| {\vec a} \right|\;\left| {\vec b} \right|, the equality holding only if \theta  = 0 or \pi
(iii) The projection of \vec a on \vec b is
{p_{ab}} = \dfrac{{\vec a \cdot \vec b}}{{\left| {\vec b} \right|}} = \vec a \cdot \left( {\dfrac{{\vec b}}{{\left| {\vec b} \right|}}} \right) = \vec a \cdot \widehat b
(iv) The projection of \vec b on \vec a is
{p_{ba}} = \dfrac{{\vec a \cdot \vec b}}{{\left| {\vec a} \right|}} = \left( {\dfrac{{\vec a}}{{\left| {\vec a} \right|}}} \right) \cdot \vec b = \widehat a \cdot \vec b
(v) Scalar product is commutative i.e.,
\vec a \cdot \vec b = \vec b \cdot \vec a
(vi) Scalar product is distributive i.e.,
\vec a \cdot \left( {\vec b + \vec c} \right) = \vec a \cdot \vec b + \vec a \cdot \vec c
and \left( {\vec a + \vec b} \right) \cdot \vec c = \vec a \cdot \vec c + \vec b \cdot \vec c
(vi) The scalar product of two vectors is zero if and only if the two vectors are perpendicular.
This also gives
\hat i \cdot \hat j = \hat j \cdot \hat i = \hat i \cdot \hat k = \hat k \cdot \hat i = \hat j \cdot \hat k = \hat k \cdot \hat j = 0
(vii) For any vector \vec a
{\vec a \cdot \vec a = {{\left| {\vec a} \right|}^2}}
Thus,
\hat i \cdot \hat i = \hat j \cdot \hat j = \hat k \cdot \hat k = 1
(viii) {\left| {\vec a \pm \vec b} \right|^2} = (\vec a \pm \vec b) \cdot (\vec a \pm \vec b)
 = {\left| {\vec a} \right|^2} + {\left| {\vec b} \right|^2} \pm 2(\vec a \cdot \vec b)
(\vec a + \vec b)(\vec a - \vec b) = {\left| {\vec a} \right|^2} - {\left| {\vec b} \right|^2}
(ix) This property is very important. If two vectors \vec a and \vec b have been specified in rectangular form
\vec a = {a_1}\,\hat i + {a_2}\,\hat j + {a_3}\,\hat k and \vec b = {b_1}\,\hat i + {b_2}\,\hat j + {b_3}\,\hat k then
We have
\vec a \cdot \vec b = \left( {{a_1}\hat i + {a_2}\hat j + {a_3}\hat k} \right)\left( {{b_1}\hat i + {b_2}\hat j + {b_3}\hat k} \right)
 = {a_1}\,{b_1} + {a_2}\,{b_2} + {a_3}\,{b_3} Using properties (vi) and (vii)
\Rightarrow\,\,\,\,{\vec a \cdot \vec b = {a_1}\,{b_1} + {a_2}\,{b_2} + {a_3}\,{b_3}}
The angle \theta  between the two vectors will be given by \cos \theta  = \dfrac{{\vec a \cdot \vec b}}{{\left| {\vec a} \right|\;\left| {\vec b} \right|}}:
 \Rightarrow  \,\,\,\, \cos \theta  = \dfrac{{{a_1}{\kern 1pt} \,{b_1} + {a_2}\,{b_2} + {a_3}\,{b_3}}}{{\sqrt {a_1^2 + a_2^2 + a_3^2} \sqrt {b_1^2 + b_2^2 + b_3^2} }}
(x) The direction cosines lmn of a vector \vec a will be given by
l = \hat a \cdot \hat i,\;\;\;m = \hat a \cdot \hat j,\;\;\;\;n = \hat a \cdot \hat k
(xi) Let \vec r be a vector coplanar with the vectors \vec a and \vec b. If \vec r \cdot \vec a = 0 and \vec r \cdot \vec b = 0, this would imply that \vec r is perpendicular to both \vec a and \vec b. This can only happen if \vec a and \vec b are collinear.
Analogously, let \vec r be an arbitrary vector and \vec a,\;\;\vec b,\;\;\vec c be three vectors such that
\vec r \cdot \vec a = \vec r \cdot \vec b = \vec r \cdot \vec c = 0
This means that \vec r is perpendicular to each of \vec a\vec b and \vec c which can only happen if \vec a,\;\vec b and \vec c are coplanar.
(xii) Let \vec a,\;\;\vec b,\;\;\vec c be three non-coplanar vectors. We’ve already discussed that \vec a,\;\;\vec b,\;\;\vec c can form a basis for 3 - D  space. Any vector \vec r can be written in this basis as
\begin{array}{l}  \vec r = (\vec r \cdot \hat a)\hat a + (\vec r \cdot \hat b)\hat b + (\vec r \cdot \hat c)\hat c\\  \,\,\,\, = \left( {\dfrac{{\vec r \cdot \vec a}}{{{{\left| {\vec a} \right|}^2}}}} \right)\vec a + \left( {\dfrac{{\vec r \cdot \vec b}}{{{{\left| {\vec b} \right|}^2}}}} \right)\vec b + \left( {\dfrac{{\vec r \cdot \vec c}}{{{{\left| {\vec c} \right|}^2}}}} \right)\vec c  \end{array}
This representation is of significant importance and you must understand how it comes about.
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