Show that the vectors  form the vertices of a right angled tri

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 Multiple Choice QuestionsLong Answer Type

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191. Show that the vectors 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 3 straight j with hat on top space minus space 5 straight k with hat on top comma space straight c with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight j with hat on top minus 4 straight k with hat on top form the vertices of a right angled triangle. 


Let 
straight a with rightwards arrow on top space equals 2 straight i with hat on top space minus 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 straight i with hat on top minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top
be the position vectors of A. B, C respectively.
therefore space space space space AB with rightwards arrow on top space equals space straight P. straight V. space of space straight B space minus space straight P. straight V. space of space straight A space equals straight b with rightwards arrow on top minus straight a with rightwards arrow on top
                              equals left parenthesis straight i with hat on top space space minus space 3 space straight j with hat on top space minus space space 5 straight k with hat on top right parenthesis space minus space left parenthesis 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top right parenthesis space equals space minus straight i with hat on top minus 2 space straight j with hat on top space minus space 6 space straight k with hat on top
         BC with rightwards arrow on top space equals space straight P. straight V. space of space straight C space minus space straight P. straight V. space of space straight B space equals space straight c with rightwards arrow on top minus straight b with rightwards arrow on top
space space space space space space space equals space left parenthesis 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space space 4 space straight k with hat on top right parenthesis space minus space left parenthesis 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 right parenthesis space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top
CA with rightwards arrow on top space equals space straight P. straight V. space of space straight A space minus space straight P. straight V. space of space straight C space equals space straight a with rightwards arrow on top minus straight c with rightwards arrow on top
space space space space space space space equals space left parenthesis 2 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 right parenthesis space minus space left parenthesis 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis space equals space minus straight i with hat on top space plus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top
therefore space space space AB with rightwards arrow on top plus BC with rightwards arrow on top space equals straight i with hat on top space space minus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top space equals space minus CA with rightwards arrow on top
therefore space space space AB with rightwards arrow on top plus BC with rightwards arrow on top plus CA with rightwards arrow on top space equals space 0 with rightwards arrow on top
Again comma space space BC with rightwards arrow on top. space CA with rightwards arrow on top space equals space left parenthesis 2 right parenthesis thin space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 5 right parenthesis space equals negative 2 minus 3 plus 5 space equals space 0
therefore space space space space BC with rightwards arrow on top space and space CA with rightwards arrow on top space are space perpendicular
therefore space space space increment ABC space is space right space angled
therefore given position vectors straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top form the vertices of a right angled triangle.

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192. Prove that the vectors
straight a with rightwards arrow on top space equals space 2 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 comma space space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space minus 3 space straight j with hat on top space minus space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 4 space straight k with hat on top forms the sides of a right-angled triangle. Also, find the remaining angles of the triangle.
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 Multiple Choice QuestionsShort Answer Type

193. Show that the points whose position vectors are 
straight a with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 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 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space space 5 space straight k with hat on top comma space space straight c with rightwards arrow on top space equals straight i with hat on top space minus space straight j with hat on top form a right angled triangle.
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194.

If the vertices A, B, C of ∆ ABC have position vectors (1, 2. 3), (– 1, 0, 0), (0, 1, 2) respectively, what is the magnitude of ∠ABC?

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195.

Show that 1 over 7 left parenthesis 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space 6 straight k with hat on top right parenthesis comma space space 1 over 7 left parenthesis 3 straight i with hat on top minus 6 straight j with hat on top plus 2 straight k with hat on top right parenthesis comma space space 1 over 7 left parenthesis 6 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top right parenthesis are mutually perpendicular unit vectors.

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 Multiple Choice QuestionsLong Answer Type

196. The scalar product of 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 with a unit vector along the sum of vectors 2 straight i with hat on top space plus space 4 space straight j with hat on top space minus space 5 straight k with hat on top and straight lambda straight i with hat on top space plus space 2 straight j with hat on top space plus space space 3 straight k with hat on top is equal to 1. Find the value of  λ.
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197. If a unit vector straight a with rightwards arrow on top makes angle straight pi over 4 space with space straight i with hat on top comma space space straight pi over 3 space with space straight j with hat on top and an acute angle straight theta with straight k with hat on top, then find the component of straight a with rightwards arrow on top and the angle straight theta.
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 Multiple Choice QuestionsShort Answer Type

198. If A, B, C have position vectors (0, 1, 1) (3, 1, 5), (0. 3, 3) respectively, then show that the ∆ABC is right angled at C.
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 Multiple Choice QuestionsLong Answer Type

199. If a unit vector straight a with rightwards arrow on top makes angles straight pi over 3 with straight i with hat on top comma  straight pi over 4 space with space straight j with hat on top and acute angle straight theta with straight k with hat on top, then find the components of straight a with rightwards arrow on top and the angle straight theta.
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200.

Let straight a with rightwards arrow on top space equals space 4 straight i with hat on top space plus space 5 straight j with hat on top space minus space straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight i with hat on top minus 4 straight j with hat on top plus space 5 straight k with hat on top space and space straight c with rightwards arrow on top space equals 3 straight i with hat on top plus straight j with hat on top minus straight k with hat on top. space Find a vector straight d with rightwards arrow on top which is perpendicular to both straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space and space straight d with rightwards arrow on top. space straight c with rightwards arrow on top space equals space 21.

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