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Vector Algebra

Question
CBSEENMA12033949
Wired Faculty App

Let straight u with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top comma space space straight v with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space and space straight w with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 3 space straight j with hat on top. Find scalars a and b such that straight w with rightwards arrow on top space equals space straight a straight u with rightwards arrow on top space plus space straight b space straight v with rightwards arrow on top.

Solution

Here,    straight u with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top comma space space straight v with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top comma space space straight w with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 3 space straight j with hat on top
therefore                                       straight w with rightwards arrow on top space equals space straight a straight u with rightwards arrow on top space plus space straight b space straight v with rightwards arrow on top.
rightwards double arrow space space space space space space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space equals space straight a space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top close parentheses space plus space straight b space open parentheses negative 2 space straight i with hat on top space plus space straight j with hat on top close parentheses rightwards double arrow space space space left parenthesis straight a minus 2 straight b minus 4 right parenthesis space straight i with hat on top space plus space left parenthesis 2 straight a plus straight b minus 3 right parenthesis space straight j with hat on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space space space space space space space space straight a minus 2 straight b minus 4 space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis rightwards double arrow space space space space space space space space 2 straight a plus straight b minus 3 space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Solving space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get space space space space space space space space space space space space space space space fraction numerator straight a over denominator 6 plus 4 end fraction space equals fraction numerator straight b over denominator negative 8 plus 3 end fraction space equals fraction numerator 1 over denominator 1 plus 4 end fraction therefore space space space space space space space space straight a over 10 space equals fraction numerator straight b over denominator negative 5 end fraction space equals 1 fifth therefore space space space space space space space space space straight a space equals space 10 over 5 space equals 2 comma space space space space straight b space equals space fraction numerator negative 5 over denominator 5 end fraction space equals space minus 1 space space

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