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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.
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