Question

If $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of 3 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 2 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 3 comma$ find the angle between $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.$

Solution

open vertical bar straight a with rightwards arrow on top close vertical bar space equals space square root of 3 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 2 comma space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space space equals 3

Let

straight theta

be angle between

straight a with rightwards arrow on top space and space straight b with rightwards arrow on top

Now,

straight a with rightwards arrow on top. space space straight b with rightwards arrow on top space equals space 3 space space space space space space space rightwards double arrow space space 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 space cos space straight theta space equals space 3

therefore space space space left parenthesis square root of 3 right parenthesis thin space left parenthesis 2 right parenthesis space cos space straight theta space equals space 3 space space space space space space rightwards double arrow space space space cos space straight theta space equals fraction numerator square root of 3 over denominator 2 end fraction therefore space space space space space space space straight theta space equals space 30 degree

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