If , respectively, are the position vectors of points A, B, C a

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 Multiple Choice QuestionsShort Answer Type

171. For any two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top comma we always have open vertical bar straight a with rightwards arrow on top. space straight b with rightwards arrow on top close vertical bar space less or equal than 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.
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172. For any two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top, we always have open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar space less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar space plus space open vertical bar straight b with rightwards arrow on top close vertical bar
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173.

If straight a with rightwards arrow on top is any vector, then show that
         straight a with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top space minus space straight i with hat on top right parenthesis space straight i with hat on top space plus space left parenthesis straight a with rightwards arrow on top. space straight j with hat on top right parenthesis space straight j with hat on top space plus space left parenthesis straight a with rightwards arrow on top space. space straight k with hat on top right parenthesis space straight k with hat on top.

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174.

If straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top comma space 2 space straight i with hat on top plus 5 space straight j with hat on top comma space space 3 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 space space and space straight i with hat on top space minus space 6 space straight j with hat on top space minus space straight k with hat on top, respectively, are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Deduce that AB and CD are parallel.


Let 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 comma space straight d with rightwards arrow on top be position vectors of points A, B, C, D respectively.
therefore space space space space 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 comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 5 space straight j with hat on top comma space straight c 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 minus space 3 space straight k with hat on top comma space straight d with rightwards arrow on top space equals space straight i with hat on top space minus space 6 space straight j with hat on top space minus space straight k with hat on top
   AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space space straight P. straight V. space of space straight A space equals space straight b with rightwards arrow on top minus straight a with rightwards arrow on top space equals space straight i with hat on top plus 4 space straight j with hat on top space minus space straight k with hat on top
CD with rightwards arrow on top space equals space straight P. straight V. space of space straight D space minus space straight P. straight V. space of space straight C space equals straight d with rightwards arrow on top space minus space straight c with rightwards arrow on top space equals space minus 2 space straight i with hat on top space minus space 8 space straight j with hat on top space plus space 2 space straight k with hat on top
therefore space space space open vertical bar AB with rightwards arrow on top close vertical bar space equals space square root of 1 plus 16 plus 1 end root space equals space square root of 18 comma space space space open vertical bar CD with rightwards arrow on top close vertical bar space equals space square root of 4 plus 64 plus 4 end root space equals square root of 72
         AB with rightwards arrow on top space. space CD with rightwards arrow on top space equals space left parenthesis 1 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis negative 8 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 2 right parenthesis space equals space minus 2 minus 32 minus 2 space equals space minus 36
Let straight theta be angle between AB and CD
therefore space space cos space straight theta space equals space fraction numerator AB with rightwards arrow on top. space CD with rightwards arrow on top over denominator open vertical bar AB with rightwards arrow on top close vertical bar space space open vertical bar CD with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator negative 36 over denominator left parenthesis square root of 18 right parenthesis space left parenthesis square root of 72 right parenthesis end fraction space equals space fraction numerator negative 36 over denominator 3 square root of 2 space cross times space 6 space square root of 2 end fraction space equals negative 36 over 36 space equals space minus 1
therefore space space space straight theta space equals space straight pi space as space 0 space less or equal than space straight theta space less or equal than space straight pi
therefore space space AB space and space CD space are space parallel.

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175. Find the projection of the vector straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 2 straight k with hat on top on the vector straight b with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top.
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176.

Find the projection of the vector straight i with hat on top space minus space straight j with hat on top on the vector straight i with hat on top space plus space straight j with hat on top.

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177.

Find the projection of the vector straight i with hat on top space plus space 3 space straight j with hat on top space plus space 7 space straight k with hat on top on the vector 7 space straight i with hat on top space minus space straight j with hat on top space plus space 8 space straight k with hat on top.

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178.

Find the projection of 8 space straight i with hat on top space plus space straight j with hat on top in the direction of straight i with hat on top space plus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top.

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179.

Find the projection of straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top in the direction of straight i with hat on top space plus space straight j with hat on top.

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180.

Find the projection of straight b with rightwards arrow on top plus straight c with rightwards arrow on top on straight a with rightwards arrow on top where
         straight a 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 space plus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top space and space straight c with rightwards arrow on top space equals space straight i with hat on top space plus space straight k with hat on top.

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