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Vector Algebra

Question
CBSEENMA12034095
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For any two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma we always have open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar.

Solution

Assume that open vertical bar straight a with rightwards arrow on top close vertical bar space not equal to space 0 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space not equal to space 0      open square brackets because space result space is space true space for space straight a with rightwards arrow on top space equals space 0 space or space straight b with rightwards arrow on top space equals space 0 close square brackets
therefore space space space fraction numerator open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar over denominator open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar space cos space straight theta over denominator open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar end fraction where space straight theta space is space angle space between space vectors therefore space space space space fraction numerator open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar over denominator open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar end fraction space equals space cos space straight theta space less or equal than space 1 therefore space space space space space open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar space less or equal than space space open vertical bar straight a with rightwards arrow on top close vertical bar space open vertical bar straight b with rightwards arrow on top close vertical bar

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