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Vector Algebra

Question
CBSEENMA12034201
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If straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are two vectors such that open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 7 and straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top space semicolon find the angle between straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.

Solution

Here straight a with rightwards arrow on top cross times straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 6 space straight k with hat on top
therefore space space open vertical bar straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 6 right parenthesis squared end root space equals square root of 9 plus 4 plus 36 end root space equals square root of 49 space equals 7 therefore space space space open vertical bar straight a with rightwards arrow on top cross times space straight b with rightwards arrow on top close vertical bar space equals space 7 therefore 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 space sin space straight theta space equals space 7 comma space space space where space straight theta space is space an space angle space between space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top therefore space space space space space left parenthesis 2 right parenthesis thin space left parenthesis 7 right parenthesis space sin space straight theta space equals space 7 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 open square brackets because space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 7 close square brackets therefore space space space sin space straight theta space equals space 1 half space space space rightwards double arrow space space space straight theta space equals space 30 degree

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