Find the angle between the vectors  and . Also find the proje

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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.


straight a with rightwards arrow on top space equals 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 comma space space space 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 space straight k with hat on top
therefore space space space space straight a space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals space square root of 3 comma space space straight b space equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared end root space equals square root of 3
straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis space equals space 1 minus 1 plus 1 space equals space 1
Let space straight theta space be space the space angle space between space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top
therefore space space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a space straight b space cos space straight theta space space space space space space space space rightwards double arrow space space space space 1 space equals space left parenthesis square root of 3 right parenthesis thin space left parenthesis square root of 3 right parenthesis space cos space straight theta space space space space space rightwards double arrow space space 1 space equals space 3 space cos space straight theta
therefore space space space space space space cos space straight theta space equals space 1 third space space space space space space space space space space space space space space space rightwards double arrow space space space space space straight theta space equals space cos space to the power of negative 1 end exponent space open parentheses 1 third close parentheses space space space
Also space projection space of space straight a with rightwards arrow on top space on space straight b with rightwards arrow on top space space equals space fraction numerator straight a with rightwards arrow on top. space straight b with rightwards arrow on top over denominator straight b end fraction space equals fraction numerator 1 over denominator square root of 3 end fraction
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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

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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