For any two vectors , we always have  from Mathematics Vector

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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


Assume that open vertical bar straight a with rightwards arrow on top close vertical bar space not equal to space 0 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space not equal to space 0 space space space space space space space left square bracket because space space result space is space true space for space straight a with rightwards arrow on top space equals space 0 space or space straight b with rightwards arrow on top space equals space 0 right square bracket
             open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar squared space equals space left parenthesis straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis squared space equals space straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared space plus space 2 space straight a with rightwards arrow on top space straight b with rightwards arrow on top
                           equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus space 2 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 cos space straight theta
less or equal than space open vertical bar straight a with rightwards arrow on top close vertical bar squared plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus space 2 space open vertical bar straight a with rightwards arrow on top close vertical bar space. space open vertical bar straight b with rightwards arrow on top close vertical bar space space space space space space space space space space space space space space space space space space space space space space left square bracket because space cos space straight theta space less or equal than space 1 right square bracket
space equals space left parenthesis 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 right parenthesis squared
therefore space space space space open vertical bar straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close vertical bar squared space space less or equal than space open parentheses 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 close parentheses squared
space space space space space space space space space open vertical bar straight a with rightwards arrow on top space plus 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 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.

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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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