Express the vector  as sum of two vectors such that one is pa

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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

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.

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 Multiple Choice QuestionsShort Answer Type

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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.


                       straight a with rightwards arrow on top space equals space 5 straight i with hat on top space minus space 2 straight j with hat on top plus 5 straight k with hat on top comma space space space straight b with rightwards arrow on top space equals 3 straight i with hat on top space plus straight k with hat on top
Let straight a with rightwards arrow on top space equals space straight A with rightwards arrow on top space plus space straight B with rightwards arrow on top space where space straight A with rightwards arrow on top space is space parallel space to space straight b with rightwards arrow on top space and space straight B with rightwards arrow on top space is space perpendicular space to space straight b with rightwards arrow on top.
therefore space space space space straight A with rightwards arrow on top space equals space straight lambda left parenthesis 3 space straight i with hat on top space plus space straight k with hat on top right parenthesis space equals space 3 space straight lambda space straight i with hat on top space plus space straight lambda space straight k with hat on top
and straight B with rightwards arrow on top space equals space straight a with rightwards arrow on top space minus space straight A with rightwards arrow on top space equals space left parenthesis 5 straight i with hat on top plus straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis space minus space left parenthesis 3 space straight lambda space straight i with hat on top space plus space straight lambda space straight k with hat on top right parenthesis
            equals space left parenthesis 5 space minus 3 space straight lambda right parenthesis space straight i with hat on top space plus space straight j with hat on top space plus space left parenthesis 5 minus straight lambda right parenthesis space straight k with hat on top
Since straight B with rightwards arrow on top is perpendicular to straight b with rightwards arrow on top
therefore space space space space straight B with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0
rightwards double arrow space space space left parenthesis 5 space minus space 3 space straight lambda right parenthesis space left parenthesis 3 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 0 right parenthesis space plus space left parenthesis 5 minus straight lambda right parenthesis space left parenthesis 1 right parenthesis space equals space 0
rightwards double arrow space space space space 15 minus 9 straight lambda plus 5 minus straight lambda space equals space 0 space space space space space space space space space space space space space space space rightwards double arrow space space 10 straight lambda space equals space 20 space space space space rightwards double arrow space space space straight lambda space equals space 2
therefore space space space space space space straight A with rightwards arrow on top space equals space 6 straight i with hat on top space plus space 2 straight k with hat on top comma space space space straight B with rightwards arrow on top space equals space minus straight i with hat on top plus space straight j with hat on top space plus space 3 straight k with hat 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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