Find the angle between vectors  and  if  and  from Mathe

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 Multiple Choice QuestionsShort Answer Type

131. Given two points A and B, identify the set of points P in space such that PA with rightwards arrow on top. space PB with rightwards arrow on top space less than space 0.
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132.

Find open vertical bar straight x with rightwards arrow on top close vertical bar comma space if for a unit vector straight a with rightwards arrow on top comma space left parenthesis straight x with rightwards arrow on top minus straight a with rightwards arrow on top right parenthesis. space left parenthesis straight x with rightwards arrow on top plus straight a with rightwards arrow on top right parenthesis space equals space 12

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133.

If straight a with rightwards arrow on top is the unit vector and open parentheses straight x with rightwards arrow on top minus straight a with rightwards arrow on top close parentheses. space left parenthesis straight x with rightwards arrow on top space plus space straight a with rightwards arrow on top right parenthesis space equals space 8 comma then find open vertical bar straight x with rightwards arrow on top close vertical bar.

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134.

If straight a with rightwards arrow on top. space straight a with rightwards arrow on top space equals space 0 and straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 comma then what can be concluded about the vector straight b with rightwards arrow on top?

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135.

If either vector straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top space space or space straight b with rightwards arrow on top space equals space 0 comma space then space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0. But the converse need not be true. Justify your answer with an example. 

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136. Find the angle between the vectors straight i with hat on top space minus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space and space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top.
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137. Find the angle between vectors straight a with rightwards arrow on top plus straight b with rightwards arrow on top and straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top if straight a 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 3 space straight k with hat on top and straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top


straight a 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 3 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top
therefore space space space space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space straight k with hat on top comma space space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space equals space minus straight i with hat on top space minus space space 2 space straight j with hat on top space plus space 5 space straight k with hat on top
space 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 equals space square root of 25 plus 1 end root space equals space square root of 26
space space space space space space space space space space space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar space equals space square root of 1 plus 4 plus 25 end root space equals space square root of 30
Let space straight theta space space be space angle space between space straight a with rightwards arrow on top plus space straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space space minus space straight b with rightwards arrow on top.
therefore space space space space space cos space straight theta space space space equals space space fraction numerator left parenthesis straight a with rightwards arrow on top space plus space 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 over denominator open vertical bar straight a with rightwards arrow on top plus space straight b with rightwards arrow on top close vertical bar. space open vertical bar straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top close vertical bar end fraction
space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator left parenthesis 5 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 26. space square root of 30 end fraction space equals space fraction numerator negative 5 plus 0 plus 5 over denominator square root of 26 space square root of 30 end fraction
therefore space space space space space cos space straight theta space equals space 0 space space space space rightwards double arrow space space space straight theta space equals space 90 degree. space
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138.

If straight a with rightwards arrow on top space equals space straight i with hat on top plus space 2 space straight j with hat on top space space minus space 3 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top comma space space show that open parentheses straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses space and space left parenthesis straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top right parenthesis are perpendicular to each other. 

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139.

If straight a with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space straight j with hat on top space minus space space 3 space straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top plus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space then show that the vectors straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top space and space straight a with rightwards arrow on top space minus space straight b with rightwards arrow on top space are space orthogonal.

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140. For what values of A are the vectors 
straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight lambda space straight j with hat on top space plus space straight k with hat on top space and space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top perpendicular to each other?
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