tgt

## 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