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## 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.