If  makes equal angles with  and has magnitude 3, then prove

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 Multiple Choice QuestionsShort Answer Type

181. Find λ when the scalar projection of straight a with rightwards arrow on top space equals space straight lambda straight i with hat on top space plus space straight j with hat on top space plus space 4 straight k with hat on top space space and space straight b with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 6 straight j with hat on top space plus space 3 straight k with hat on top space is space 4 space units.
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182. Find the angle between the vectors straight a with rightwards arrow on top space equals straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space and straight b with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top.
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183. Find the angle between the vectors straight a with rightwards arrow on top space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top and straight b with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus straight k with hat on top. Also find the projection of straight a with rightwards arrow on top space on space straight b with rightwards arrow on top.
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184. Find the cosine of the angle between the vectors straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space space and space straight b with rightwards arrow on top space equals 4 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top
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185. Find the cosine of the angle between the vectors
straight a with rightwards arrow on top space equals space 5 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 space and space straight b with rightwards arrow on top space equals space minus 2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top
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 Multiple Choice QuestionsLong Answer Type

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186.

If straight a with rightwards arrow on top makes equal angles with straight i with hat on top comma space straight j with hat on top space and space straight k with hat on top and has magnitude 3, then prove that the angle between straight a with rightwards arrow on top and each of straight i with hat on top comma space straight j with hat on top space and space straight k with hat on top is cos to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses.


Let straight a with rightwards arrow on top space equals straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top such that open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3 and O be the angle which straight a with rightwards arrow on top makes with straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top.
therefore space space space space cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight i with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight i with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 1 over denominator left parenthesis 3 right parenthesis left parenthesis 1 right parenthesis end fraction space space space rightwards double arrow space space space space straight a subscript 1 space equals space 3 space cos space straight theta

Again cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight j with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight j with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 2 over denominator left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis end fraction space space space rightwards double arrow space space straight a subscript 2 space equals space 3 space cos space straight theta

and cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight k with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight j with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 3 over denominator left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis space end fraction space space rightwards double arrow space space straight a subscript 3 space equals space 3 space cos space straight theta

Also, open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3                rightwards double arrow space space space space space square root of straight a subscript 1 squared plus straight a subscript 2 squared plus straight a subscript 3 squared end root space equals space 3

rightwards double arrow space space space square root of 9 space cos squared straight theta space plus space 9 space cos squared straight theta space plus space 9 space cos squared space straight theta end root space equals space 3 space space space space space space space rightwards double arrow space space space square root of 27 space cos squared straight theta end root space equals space 3
rightwards double arrow space space space 3 square root of 3 space cos space straight theta space equals space 3 space space space space space space rightwards double arrow space space space cos space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction
therefore space space space space space space space straight theta space space equals space cos to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses


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 Multiple Choice QuestionsShort Answer Type

187. Express the vector straight a with rightwards arrow on top space equals space 5 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top as sum of two vectors such that one is parallel to the vector straight b with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight k with hat on top and other is perpendicular to straight b with rightwards arrow on top.
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188. Decompose the vector 6 straight i with hat on top space minus space 3 straight j with hat on top space minus space 6 straight k with hat on top into vectors which are parallel and perpendicular to the vector straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top.
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 Multiple Choice QuestionsLong Answer Type

189. Express 2 straight i with hat on top space minus straight j with hat on top space plus space 3 straight k with hat on top as the sum of a vector parallel, and a vector parallel, and a vector perpendicular to 2 straight i with hat on top plus 4 straight j with hat on top minus 2 straight k with hat on top.
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 Multiple Choice QuestionsShort Answer Type

190. Show that the vectors straight a with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space straight b with rightwards arrow on top space equals 2 space straight i with hat on top space minus straight j with hat on top plus straight k with hat on top comma space and space straight c with rightwards arrow on top space equals straight i with hat on top plus 2 space straight j with hat on top plus 6 space straight k with hat on top form a right angled triangle. 
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