Decompose the vector  into vectors which are parallel and perp

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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

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.


Let straight a with rightwards arrow on top space equals space 6 straight i with hat on top space minus space 3 straight j with hat on top minus space 6 straight k with hat on top be decomposed into two vectors straight A with rightwards arrow on top space and space straight B with rightwards arrow on top.
where straight A with rightwards arrow on top is parallel to 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 plus space straight k with hat 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 straight A with rightwards arrow on top space equals space straight lambda space left parenthesis straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top right parenthesis space equals space straight lambda space straight i with hat on top space plus space straight lambda space straight j with hat on top space plus space straight lambda space straight k with hat on top
and     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 space space space space space space space space space space space space space space space space rightwards double arrow space space space space 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
therefore space space space straight B with rightwards arrow on top space equals space left parenthesis 6 minus space straight lambda right parenthesis space straight i with hat on top space minus space left parenthesis straight lambda plus 3 right parenthesis space straight j with hat on top space minus space left parenthesis straight lambda minus 6 right parenthesis space straight k with hat on top
because space 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 b with rightwards arrow on top. space straight B with rightwards arrow on top space equals space 0
rightwards double arrow space space left parenthesis 6 minus straight lambda right parenthesis space left parenthesis 1 right parenthesis space minus space left parenthesis straight lambda plus 3 right parenthesis space left parenthesis 1 right parenthesis space minus space left parenthesis straight lambda plus 6 right parenthesis space left parenthesis 1 right parenthesis space equals space 0
therefore space space space space 6 space minus space straight lambda space minus straight lambda minus 3 minus straight lambda minus 6 space equals space 0 space space space space space space space rightwards double arrow space space space space space 3 space straight lambda space equals space minus 3 space space space space rightwards double arrow space space space straight lambda space equals space minus 1
therefore space space space required space vectors space are
space space space space space space space space space space space space space minus straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top comma space space space space space 7 straight i with hat on top space minus space 2 space straight j with hat on top space minus space 5 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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