Find the angle between the vectors  and  from Mathematics Ve

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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.


Let straight theta be angle between the vectors
                    straight a 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 space and space 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
Now,      straight a space equals 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 space square root of 1 plus 1 plus 1 end root space equals square root of 3
              straight b equals space square root of left parenthesis 1 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis negative 1 right parenthesis squared end root space equals square root of 1 plus 1 plus 1 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 negative 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 equals space 1 minus 1 minus 1 space equals negative 1
Now    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 rightwards double arrow space space space space minus 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
therefore space space space space space 3 space cos space straight theta space equals space minus 1 space space space space space space space space space or space space space cos space straight theta space equals space minus 1 third
therefore space space space space space space space space space straight theta space equals space cos to the power of negative 1 end exponent space open parentheses negative 1 third close parentheses.
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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

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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