ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 4 Worked Out Examples

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

Friday, 8 August 2014

CHAPTER 4 Worked Out Examples

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