Find  and  if   from Mathematics Vector Algebra

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 Multiple Choice QuestionsShort Answer Type

151. 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 2 comma space 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 3 comma find the angle between straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.
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152.

Find the angle between two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top having the same length (magnitude) square root of 2 and their scalar products is -1.

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153.

Find the angle between two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top such that open vertical bar straight a with rightwards arrow on top close vertical bar space equals space open vertical bar straight b with rightwards arrow on top close vertical bar space equals space 3 and straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 1.

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154. Find the magnitude of two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top having the same magnitude and such that the angle between them is 30° and their scalar product is 3.
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155. Find the magnitude of two vectors straight a with rightwards arrow on top space and space straight b with rightwards arrow on top having the same magnitude and such that the angle between then is 60° and their scalar product is 1/2.
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156.

Find open vertical bar straight a with rightwards arrow on top close vertical bar and open vertical bar straight b with rightwards arrow on top close vertical bar comma if  open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses. space left parenthesis straight a with rightwards arrow on top space minus straight b with rightwards arrow on top right parenthesis space equals space 12 space and space open vertical bar straight a with rightwards arrow on top close vertical bar space equals 2 space open vertical bar straight b with rightwards arrow on top close vertical bar.


We have
        left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space equals space 12
therefore space space space space open vertical bar straight a with rightwards arrow on top close vertical bar squared minus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space 12
rightwards double arrow space space space space left parenthesis 2 space open vertical bar straight b with rightwards arrow on top close vertical bar right parenthesis squared space minus space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space 12 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 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 space open vertical bar straight b with rightwards arrow on top close vertical bar close square brackets
rightwards double arrow space space space space 4 space open vertical bar straight b with rightwards arrow on top close vertical bar squared space minus open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals 12 space space space space space space space space rightwards double arrow space space 3 space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space 12
therefore space space space space open vertical bar straight b with rightwards arrow on top close vertical bar squared space equals space 4 space space space space space space space space space space space space space space space space space rightwards double arrow space space space space open vertical bar straight b with rightwards arrow on top close vertical bar space space equals space 2
therefore space space space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 space cross times 2 space equals space 4 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 space space space space space space space space space space space space open square brackets because space space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 2 space open vertical bar straight b with rightwards arrow on top close vertical bar close square brackets space space

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157.

Find open vertical bar straight a with rightwards arrow on top close vertical bar space and space space open vertical bar straight b with rightwards arrow on top close vertical bar space if space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis space equals space 8 space space and space open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 8 space open vertical bar straight b with rightwards arrow on top close vertical bar.

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158.

Find open vertical bar straight a with rightwards arrow on top close vertical bar space and space open vertical bar straight b with rightwards arrow on top close vertical bar space if space space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. thin space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space equals 3 space and space 2 space open vertical bar straight b with rightwards arrow on top close vertical bar space equals open vertical bar straight a with rightwards arrow on top close vertical bar.

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159.

If left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. space left parenthesis straight a with rightwards arrow on top minus straight b with rightwards arrow on top right parenthesis space equals 0 show that open vertical bar straight a with rightwards arrow on top close vertical bar space equals open vertical bar straight b with rightwards arrow on top close vertical bar

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160.

If 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 comma spaceprove that either straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top or straight b with rightwards arrow on top space equals space straight c with rightwards arrow on top  or  straight a with rightwards arrow on top  is perpendicular to straight b with rightwards arrow on top minus straight c with rightwards arrow on top.

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