If  such that each is perpendicular to sum of the other two, f

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 Multiple Choice QuestionsShort Answer Type

161.

Prove that open vertical bar straight a with rightwards arrow on top close vertical bar space straight b with rightwards arrow on top space plus space open vertical bar straight b with rightwards arrow on top close vertical bar space straight a with rightwards arrow on top is orthogonal to open vertical bar straight a with rightwards arrow on top close vertical bar space straight b with rightwards arrow on top space minus space open vertical bar straight b with rightwards arrow on top close vertical bar space straight a with rightwards arrow on top, for any vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.

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162.

Prove that
left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses space equals 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
if and only if straight a with rightwards arrow on top comma space straight b with rightwards arrow on top are orthogonal.

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163.

Let straight a with rightwards arrow on top space and space straight b with rightwards arrow on top be the two non-zero vectors. Then, prove that straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular if and only if open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top close vertical bar space equals space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar.

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164.

Prove that two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular if 
       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 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

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165.

If straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are perpendicular, then left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis squared space equals space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis squared
 

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 Multiple Choice QuestionsLong Answer Type

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166.

If open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3 comma space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 space and space open vertical bar straight c with rightwards arrow on top close vertical bar space equals 5 such that each is perpendicular to sum of the other two, find open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar.


We have
        open vertical bar straight a with rightwards arrow on top close vertical bar space equals 3 comma space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 comma space space open vertical bar straight c with rightwards arrow on top close vertical bar space equals space 5                            ...(1)
 Since straight a with rightwards arrow on top is perpendicular to straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top comma space straight b with rightwards arrow on top space is space perp comma space straight c with rightwards arrow on top plus straight a with rightwards arrow on top space and space straight c with rightwards arrow on top space is space perp. space to space straight a with rightwards arrow on top plus straight b with rightwards arrow on top
therefore space space space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space straight b with rightwards arrow on top. space space space left parenthesis straight c with rightwards arrow on top space plus space straight a with rightwards arrow on top right parenthesis space equals space straight c with rightwards arrow on top. space left parenthesis straight a with rightwards arrow on top plus space straight b with rightwards arrow on top right parenthesis space equals space 0
therefore space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space straight b with rightwards arrow on top. space straight c with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space equals space straight c with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
Now,  open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar squared space equals space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis                                
        equals space straight a with rightwards arrow on top. straight a with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. straight c with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight b with rightwards arrow on top space. straight c with rightwards arrow on top space plus space straight c with rightwards arrow on top. straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight c with rightwards arrow on top
space equals space open vertical bar straight a with rightwards arrow on top close vertical bar squared thin space left parenthesis straight a with rightwards arrow on top. space straight b with rightwards arrow on top space plus space straight a with rightwards arrow on top. space straight c with rightwards arrow on top right parenthesis space plus space left parenthesis straight b with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top. space straight c with rightwards arrow on top right parenthesis space plus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space plus left parenthesis straight c with rightwards arrow on top. space straight a with rightwards arrow on top space plus space straight c with rightwards arrow on top. space straight b with rightwards arrow on top right parenthesis space plus space open vertical bar straight c with rightwards arrow on top close vertical bar squared
space equals left parenthesis 3 right parenthesis squared plus 0 plus 0 plus left parenthesis 4 right parenthesis squared plus 0 plus left parenthesis 5 right parenthesis squared 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 of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis close square brackets
equals space 9 plus 16 plus 25 space equals space 50
therefore space space space space open vertical bar straight a with rightwards arrow on top plus straight b with rightwards arrow on top plus straight c with rightwards arrow on top close vertical bar space equals space square root of 50 space equals space 5 square root of 2

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 Multiple Choice QuestionsShort Answer Type

167.

Find open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar comma space if two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are such that 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 3 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 4.

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168.

Find open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar comma if 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 5 space space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 8

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169.

Find open vertical bar straight a with rightwards arrow on top minus straight b with rightwards arrow on top close vertical bar squared if open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3 comma space space space space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 4 space and space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1.

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170.

If straight a with rightwards arrow on top space and space straight b with rightwards arrow on top are unit vectors, and straight theta is the angle between them, find 1 half open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar in terms of straight theta.

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