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Vector Algebra

Question
CBSEENMA12033986
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If A is (a1 , a2 , a3), AB with rightwards arrow on top is vector straight b with rightwards arrow on top space equals space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top, find the coordinates of B.

Solution

Here straight A space equals space left parenthesis straight a subscript 1 comma space straight a subscript 2 comma space straight a subscript 3 right parenthesis space space rightwards double arrow space space space space space space straight P. straight V. space space of space straight A space equals space straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top
      Let B be left parenthesis straight lambda subscript 1 comma space straight lambda subscript 2 comma space straight lambda subscript 3 right parenthesis
therefore space space space space space straight P. straight V. space of space straight B space equals space straight lambda subscript 1 straight i with hat on top space plus space straight lambda subscript 2 straight j with hat on top space plus space straight lambda subscript 3 straight k with hat on top
Now,   AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A
rightwards double arrow space space space straight b with rightwards arrow on top space equals space left parenthesis straight lambda subscript 1 straight i with hat on top space plus space straight lambda subscript 2 straight j with hat on top space plus space straight lambda subscript 3 straight k with hat on top right parenthesis space minus space left parenthesis straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top right parenthesis rightwards double arrow space space space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top space equals space left parenthesis straight lambda subscript 1 minus straight a subscript 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight lambda subscript 2 minus straight a subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight lambda subscript 3 minus straight a subscript 3 right parenthesis space straight k with hat on top rightwards double arrow space space space straight lambda subscript 1 minus straight a subscript 1 space equals space straight b subscript 1 comma space straight lambda subscript 2 minus straight a subscript 2 space equals space straight b subscript 2 comma space space space space straight lambda subscript 3 minus straight a subscript 3 space equals space straight b subscript 3 rightwards double arrow space space space space straight lambda subscript 1 space equals space straight a subscript 1 space plus space straight b subscript 1 comma space space straight lambda subscript 2 space equals space straight a subscript 2 space plus space straight b subscript 2 comma space space straight lambda subscript 3 space equals space straight a subscript 3 space plus space straight b subscript 3 therefore space space space straight B space is space left parenthesis straight a subscript 1 plus straight b subscript 1 comma space straight a subscript 2 plus straight b subscript 2 comma space straight a subscript 3 plus straight b subscript 3 right parenthesis.

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