Friday, 8 August 2014

CHAPTER 10 - Vector Terminology


Consider a vector
\vec r = x\hat i + y\hat j + z\hat k
as shown in the figure below:
The magnitude or \vec r is simply the length of the diagonal of the cuboid whose sides are xy and z. Thus
\left| {\vec r} \right| = \sqrt {{x^2} + {y^2} + {z^2}} \ldots(1)
Suppose \vec r makes angles \alpha ,\beta  and \lambda  with the XY and Z axis, as shown:
Then the quantities
l = \cos \alpha
m = \cos \beta
n = \cos \gamma
are called the direction cosines of \vec r (abbreviated as DCs. The DCs uniquely determine the direction of the vector. Note that since
\vec r = x\hat i + y\hat j + z\hat k
We have
x = \left| {\vec r} \right|\cos \alpha  = l\left| {\vec r} \right|
y = \left| {\vec r} \right|\cos \beta  = m\left| {\vec r} \right|
z = \left| {\vec r} \right|\cos \gamma  = n\left| {\vec r} \right|
 \Rightarrow  \,\,\,\, {x^2} + {y^2} + {z^2} = \left( {{l^2} + {m^2} + {n^2}} \right){\left| {\vec r} \right|^2}
From (1), this gives
{{l^2} + {m^2} + {n^2} = 1}
We can also infer from this discussion that the unit vector \hat r along \vec r can be written as
\hat r = \dfrac{{\vec r}}{{\left| {\vec r} \right|}} = \dfrac{{x\hat i + y\hat j + z\hat k}}{{\left| {\vec r} \right|}}
 = l\hat i + y\hat j + z\hat k
Direction ratios (DRs) of a vector are simply three numbers, say ab and c, which are proportional to the DCs, i.e
\dfrac{l}{a} = \dfrac{m}{b} = \dfrac{n}{c}
It follows that DRs are not uniqe (DCs) obviously are)
From a set of DRs {abc}, the DCs can easily be deduced:
\dfrac{l}{a} = \dfrac{m}{b} = \dfrac{n}{c} = \dfrac{{\sqrt {{l^2} + {m^2} + {n^2}} }}{{\sqrt {{a^2} + {b^2} + {c^2}} }} = \dfrac{1}{{\sqrt {{a^2} + {b^2} + {c^2}} }}
 \Rightarrow  \,\,\,\,\, l = \dfrac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},m = \dfrac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }},n = \dfrac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}
Before we go on to solving examples involving the concepts we’ve seen till now, you are urged to once again go over the entire earlier discussion we’ve had, so that the “big picture” is clear in your mind.
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