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Vector Algebra

Question
CBSEENMA12033952
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If straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are the vectors determined by two adjacent sides of a regular hexagon, find the vectors determined by the other sides taken in order.

Solution
Let ABCDHF be a regular hexagon such that
                         AB with rightwards arrow on top space equals space space straight a with rightwards arrow on top comma space space space BC with rightwards arrow on top space equals space straight b with rightwards arrow on top
Now,           AC with rightwards arrow on top equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top
                     AD with rightwards arrow on top space equals space 2 space BC with rightwards arrow on top space space space space space space space space space space space space space space space space space space space space space space open square brackets because space AD space equals space 2 space BC space and space AD thin space vertical line vertical line thin space BC close square brackets

therefore space space space space space space space AD with rightwards arrow on top space equals space 2 space straight b with rightwards arrow on top
           In increment ADC. space space AC with rightwards arrow on top space plus space CD with rightwards arrow on top space equals space AD with rightwards arrow on top
      therefore space space space space space CD with rightwards arrow on top space equals space AD with rightwards arrow on top space minus space AC with rightwards arrow on top space equals space 2 straight b with rightwards arrow on top space minus space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis space equals space straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top
              DE with rightwards arrow on top space equals space minus AB with rightwards arrow on top space equals space minus straight a with rightwards arrow on top EF with rightwards arrow on top space equals space minus BC with rightwards arrow on top space equals space minus straight b with rightwards arrow on top FA with rightwards arrow on top space equals space minus CD with rightwards arrow on top space equals space minus left parenthesis straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top right parenthesis space equals space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top
          

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