ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 5 - Worked Out Examples

Friday 8 August 2014

CHAPTER 5 - Worked Out Examples – 2

Example: 2

Suppose that the vectors $\vec a\,$ and $\vec b$ represent two adjacent sides of a regular hexagon. Find the vectors representing the other sides.

Solution: 2

Let the hexagon be ${A_1}{A_2}{A_3}{A_4}{A_5}{A_6}$ , as shown:

First of all, we note an important geometrical property of a regular hexagon:

	Diagonal $=2\times$ side
	$\Rightarrow \,\,\,\,\,{A_1}{A_4} = 2 \times {A_2}{A_3}$

Also, since ${A_1}{A_4}||{A_2}{A_3}$ , we have

	$\overrightarrow {{A_1}{A_4}} = 2 \times \overrightarrow {{A_2}{A_3}}$
	$= 2\,\vec b$

Now we use the triangle law to determine the various sides:

	$\overrightarrow {{A_3}{A_4}} = \overrightarrow {{A_1}{A_4}} - \overrightarrow {{A_1}{A_3}}$
	$= 2\vec b - \left( {\vec a + \vec b} \right)$
	$= \vec b - \vec a$
	$\overrightarrow {{A_4}{A_5}} = - \vec a$ (only the sense differs; support is parallel to the support of $\vec a$ )
	$\overrightarrow {{A_5}{A_6}} = - \,\overrightarrow {{A_2}{A_3}}$
	$= - \vec b$
	$\overrightarrow {{A_6}{A_1}} = - \,\overrightarrow {{A_3}{A_4}}$
	$= \vec a - \vec b$

Thus, all sides are expressible in terms of $\vec a$ and $\vec b$ .

Example: 3

What can be interpreted about $\vec a$ and $\vec b$ if they satisfy the relation:

$\left| {\vec a + \vec b} \right| = \left| {\vec a - \vec b} \right|$

Solution: 3

Make $\vec a$ and $\vec b$ co-initial so that they form the adjacent sides of a parallelogram:

We have,

	$\left\| {\vec a + \vec b} \right\| = \left\| {\overrightarrow {OC} } \right\| = OC$
	and $\left\| {\vec a - \vec b} \right\| = \left\| {\overrightarrow {BA} } \right\| = BA$

Thus, the stated relation implies that the two diagonals of the parallelogram $OACB$ are equal, which can only happen if $OACB$ is a rectangle.

This implies that $\vec a$ and $\vec b$ form the adjacent sides of a rectangle. In other words, $\vec a$ and $\vec b$ are perpendicular to each other.

ONE MORE STEP TOWARD BETTER TOMORROW

Friday 8 August 2014

CHAPTER 5 - Worked Out Examples – 2

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