Find the cosine of the angle between the vectors from Mathemati

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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


Here,   straight a with rightwards arrow on top space equals space 5 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 negative 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
therefore space space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space left parenthesis 5 straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis. space left parenthesis negative 2 straight i with hat on top space plus straight j with hat on top plus 3 space straight k with hat on top right parenthesis
space space space space space space space space space space space space space space space space space equals space left parenthesis 5 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus 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 3 right parenthesis space equals negative 10 plus 1 plus 3 space equals space minus 6
Let straight theta be angle between straight a with rightwards arrow on top space and space straight b with rightwards arrow on top.
therefore 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

   rightwards double arrow space space space minus 6 space equals space square root of 5 plus 1 plus 1 end root space square root of 4 plus 1 plus 9 end root space cos space straight theta
rightwards double arrow space space minus 6 space equals space square root of 7 space square root of 14 space cos space straight theta space space rightwards double arrow space space space space minus 6 space equals space 7 square root of 2 space cos space straight theta
therefore space space space cos space straight theta space equals space minus fraction numerator 6 over denominator 7 square root of 2 end fraction

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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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