Friday, 8 August 2014

CHAPTER 4 Worked Out Examples

  Example: 1    

From any two vectors \vec a and \vec b, prove that
(i) \left| {\,\vec a + \vec b\,} \right| \le \left| {\,\vec a\,} \right| + \left| {\,\vec b\,} \right|
(ii) \left| {\,\vec a - \vec b\,} \right| \le \left| {\,\vec a\,} \right| + \left| {\,\vec b\,} \right|
(iii) \left| {\,\vec a + \vec b\,} \right| \ge \left| {\left| {\,\vec a\,} \right| - \left| {\,\vec b\,} \right|} \right|
When does the equality hold in these cases?
Solution: 1

Consider this figure:
The first two relations follow from the fact that in any triangle, the sum of two sides is greater than the third side:
In \Delta ABC
AC \le AB + BC (we,ll soon talk about how and when the equality comes)
 \Rightarrow  \,\,\,\, \left| {\vec a + \vec b} \right| \le \left| {\vec a} \right| + \left| {\vec b} \right|
In \Delta ABC'
AC' \le AB + BC' = AB + BC
 \Rightarrow  \,\,\,\, \left| {\vec a - \vec b} \right| \le \left| {\vec a} \right| + \left| {\vec b} \right|
In the first relation, the equality can hold only if the two vectors have the same direction; this should be intuitively obvious:
The equality in the second relation holds if the two vectors are exactly opposite
To prove the third relation, we use in \Delta ABC in Fig - 11, the geometrical fact that the difference of any two sides of a triangle is less than its third side:
\left| {AB - BC} \right| \le AC
 \Rightarrow  \,\,\,\,\, \left| {\left| {\vec a} \right| - \left| {\vec b} \right|} \right| \le \left| {\vec a + \vec b} \right|
The equality holds when \vec a and \vec b are precisely in the opposite direction
The main point to understand from this example is how easily vector relations follows from corresponding geometrical facts
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