being a right-handed orthogonal system of unit vectors in spa

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 Multiple Choice QuestionsLong Answer Type

291.

If straight a with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space and straight b with rightwards arrow on top space equals space straight j with hat on top space minus space straight k with hat on top.  Find a vector straight c with rightwards arrow on top such that straight a with rightwards arrow on top cross times space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top and straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 3.

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292.

Let straight a with rightwards arrow on top space equals straight i with hat on top space plus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top comma space space space space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space space 7 space straight k with hat on top and straight c with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 4 space straight k with hat on top. Find a vector straight d with rightwards arrow on top which is perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma space space straight c with rightwards arrow on top. space straight d with rightwards arrow on top space equals space 15.

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293. If with reference to a right handed system of mutually perpendicular unit vector unit vector straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top

express straight beta with rightwards arrow on top in the form straight beta with rightwards arrow on top in the form straight beta with rightwards arrow on top space equals space straight beta with rightwards arrow on top subscript 1 space equals space straight beta with rightwards arrow on top subscript 1 space plus space straight beta with rightwards arrow on top subscript 2 space end subscript space where space stack straight beta subscript 1 with rightwards arrow on top space is space parallel space to space straight alpha space and space stack straight beta subscript 2 with rightwards arrow on top space is space perpendicular space to space straight alpha with rightwards arrow on top.
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 Multiple Choice QuestionsShort Answer Type

294. straight A with rightwards arrow on top comma space straight B with rightwards arrow on top comma space straight C with rightwards arrow on top are unit vectors,  Suppose that  straight A with rightwards arrow on top. space straight B with rightwards arrow on top space equals space straight A with rightwards arrow on top. space straight C with rightwards arrow on top space equals space 0 and the angle between B and C is straight pi over 6. Prove that straight A with rightwards arrow on top space equals plus-or-minus 2 space open parentheses straight B with rightwards arrow on top space cross times space straight C with rightwards arrow on top close parentheses.
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 Multiple Choice QuestionsLong Answer Type

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295. Give space space space straight a with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis comma space space straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis comma
space space space space space space space space space space space space space space straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis comma
straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top being a right-handed orthogonal system of unit vectors in space. Show that straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top in another such system.


        straight a with rightwards arrow on top space equals space 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 6 straight k with hat on top right parenthesis space equals 2 over 7 straight i with hat on top plus 3 over 7 straight j with hat on top space plus space 6 over 7 straight k with hat on top
straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 3 over 7 straight i with hat on top space minus space 6 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top
straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 straight i with hat on top space plus space 2 straight j with hat on top space minus space 3 straight k with hat on top right parenthesis space equals space 6 over 7 straight i with hat on top space plus space 2 over 7 straight j with hat on top space minus space 3 over 7 straight k with hat on top

therefore space space space space straight a space equals space square root of open parentheses 2 over 7 close parentheses squared plus open parentheses 3 over 7 close parentheses squared plus open parentheses 6 over 7 close parentheses squared end root space equals space square root of 4 over 49 plus 9 over 49 plus 36 over 49 end root space equals square root of 49 over 49 end root space equals 1

          straight b equals space square root of open parentheses 3 over 7 close parentheses squared plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses squared plus open parentheses 2 over 7 close parentheses squared end root space equals space square root of 9 over 49 plus 36 over 49 plus 4 over 49 end root space equals space square root of 49 over 49 end root space equals 1
straight c space equals space square root of open parentheses 6 over 7 close parentheses squared plus open parentheses 2 over 7 close parentheses squared plus open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses squared end root space equals square root of 36 over 49 plus 4 over 49 plus 9 over 49 end root space equals square root of 49 over 49 end root space equals space 1
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top space are space unit space vectors.
                straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 3 over 7 close parentheses plus space open parentheses 3 over 7 close parentheses space open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space plus space open parentheses 6 over 7 close parentheses space open parentheses 2 over 7 close parentheses space equals 6 over 49 minus 18 over 49 plus 12 over 49 equals 0
straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 6 over 7 close parentheses space plus space open parentheses 3 over 7 close parentheses space open parentheses 2 over 7 close parentheses space plus open parentheses 6 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals 12 over 49 plus 6 over 49 minus 18 over 49 space equals 0
straight b with rightwards arrow on top. straight c with rightwards arrow on top space equals space open parentheses 3 over 7 close parentheses space open parentheses 6 over 7 close parentheses plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space open parentheses 2 over 7 close parentheses plus open parentheses 2 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals space 18 over 49 minus 12 over 49 minus 6 over 49 equals space 0
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top are mutually orthogonal vectors.
Now,  
   straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 1 over 49 open vertical bar table row cell straight i with hat on top end cell cell space straight j with hat on top end cell cell straight k with hat on top end cell row 2 cell space space 3 end cell 6 row 3 cell negative 6 end cell 2 end table close vertical bar space equals space 1 over 49 open square brackets left parenthesis 6 plus 36 right parenthesis space straight i with hat on top space minus space left parenthesis 4 minus 18 right parenthesis space straight j with hat on top space plus space left parenthesis negative 12 minus 9 right parenthesis straight k with hat on top close square brackets
space space space space space space space space space space space space space equals space 1 over 49 left square bracket 42 straight i with hat on top space plus space 14 straight j with hat on top space minus space 21 straight k with hat on top right square bracket space equals space 1 over 7 left parenthesis 6 straight i with hat on top space plus 2 straight j with hat on top space minus 3 straight k with hat on top right parenthesis space equals space straight c with rightwards arrow on top
straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space 1 over 49 open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 3 cell negative 6 end cell 2 row 6 2 cell negative 3 end cell end table close vertical bar space equals 1 over 49 open square brackets left parenthesis 18 minus 4 right parenthesis straight i with hat on top space minus space left parenthesis negative 9 minus 12 right parenthesis space straight j with hat on top space plus space left parenthesis 6 plus 36 right parenthesis space straight k with hat on top close square brackets
space space space space space space space space space space space space space space equals space 1 over 49 left square bracket 14 space straight i with hat on top space plus 21 space straight j with hat on top space plus space 42 space straight k with hat on top right square bracket space equals 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus 6 space straight k with hat on top right parenthesis space equals space straight a with rightwards arrow on top
straight c with rightwards arrow on top cross times space straight a with rightwards arrow on top space equals 1 over 49 open vertical bar table row cell straight i with hat on top end cell cell space straight j with hat on top end cell cell straight k with hat on top end cell row 6 cell space space 2 end cell cell negative 3 end cell row 2 cell space space 3 end cell cell space space 6 end cell end table close vertical bar space equals space 1 over 49 open square brackets left parenthesis 12 plus 9 right parenthesis space straight i with hat on top space minus space left parenthesis 36 plus 6 right parenthesis space straight j with hat on top space plus space left parenthesis 18 minus 4 right parenthesis straight k with hat on top close square brackets space
space space space space space space space space space space space space equals space 1 over 49 left square bracket 21 space straight i with hat on top space minus space 42 space straight j with hat on top space plus space 14 space straight k with hat on top right square bracket space equals space 1 over 7 left parenthesis 3 space straight i with hat on top minus 6 space straight j with hat on top space plus 2 space straight k with hat on top right parenthesis space equals straight b with rightwards arrow on top
therefore space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top from a right handed orthogonal system of unit vectors. 





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 Multiple Choice QuestionsShort Answer Type

296. Find the value of the following:
straight i with hat on top. space open parentheses straight j with hat on top space cross times space straight k with hat on top close parentheses space plus space straight j with hat on top. space space left parenthesis straight i with hat on top space cross times space straight k with hat on top right parenthesis space plus space space straight k with hat on top. space space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis
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297. Find the value of the following:
straight i with hat on top. space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight j with hat on top space cross times space left parenthesis straight k with hat on top space cross times space straight i with hat on top right parenthesis space plus space straight k with hat on top space cross times space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis
space
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298. Find the value of the following:
straight i with hat on top space cross times space left parenthesis straight j with hat on top space cross times space straight k with hat on top right parenthesis space plus space straight j with hat on top space cross times space left parenthesis straight k with hat on top space cross times space straight i with hat on top right parenthesis space plus space straight k with hat on top space cross times space left parenthesis straight i with hat on top space cross times space straight j with hat on top right parenthesis
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 Multiple Choice QuestionsMultiple Choice Questions

299.

Let vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top be such that open vertical bar straight a with rightwards arrow on top close vertical bar space equals 3 space space space and space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space fraction numerator square root of 2 over denominator 3 end fraction comma then straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top is a unit vector, if the angle betweeen straight a with rightwards arrow on top space and space straight b with rightwards arrow on top is 

  • straight pi divided by 6
  • straight pi divided by 4
  • straight pi divided by 3
  • straight pi divided by 3
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300.

If straight theta is the angle between any two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma then open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar space equals 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 when straight theta is equal to 

  • 0

  • straight pi over 4
  • straight pi over 2
  • straight pi over 2
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