Show that  are mutually perpendicular unit vectors. from Mat

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 Multiple Choice QuestionsLong Answer Type

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. 

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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.


Let straight a with rightwards arrow on top space equals space 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 space equals space 2 over 7 straight i with hat on top space plus space 3 over 7 straight j with hat on top space plus space 6 over 7 straight k with hat on top      
straight b with rightwards arrow on top space equals space 1 over 7 left parenthesis 3 straight i with hat on top minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals 3 over 7 straight i with hat on top space minus space 6 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top
straight c with rightwards arrow on top space equals space 1 over 7 left parenthesis 6 straight i with hat on top plus space 2 straight j with hat on top space minus space 3 straight k with hat on top right parenthesis space equals space 6 over 7 straight i with hat on top plus 2 over 7 straight j with hat on top minus 3 over 7 straight k with hat on top
therefore space straight a space equals space square root of open parentheses 2 over 7 close parentheses squared plus space open parentheses 3 over 7 close parentheses squared plus open parentheses 6 over 7 close parentheses squared end root space equals space square root of 4 over 49 plus 9 over 49 plus 36 over 49 end root space equals space square root of 49 over 49 end root space equals space 1
     straight b space equals space square root of open parentheses 3 over 7 close parentheses squared plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses squared plus open parentheses 2 over 7 close parentheses squared end root space equals space square root of 9 over 49 plus 36 over 49 plus 4 over 49 end root space equals square root of 49 over 49 end root space equals space 1
straight c space equals space square root of open parentheses 6 over 7 close parentheses squared plus open parentheses 2 over 7 close parentheses squared plus open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses squared end root space equals square root of 36 over 49 plus 4 over 49 plus 9 over 49 end root space equals space square root of 49 over 49 end root space equals space 1
therefore space space 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 space are space unit space vectors.
straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 3 over 7 close parentheses space plus space open parentheses 3 over 7 close parentheses space open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space plus space open parentheses 6 over 7 close parentheses space open parentheses 2 over 7 close parentheses space equals space 6 over 49 minus 18 over 49 plus 12 over 49 space equals space 0
straight a with rightwards arrow on top. space straight c with rightwards arrow on top space equals space open parentheses 2 over 7 close parentheses space open parentheses 6 over 7 close parentheses plus open parentheses 3 over 7 close parentheses open parentheses 2 over 7 close parentheses plus open parentheses 6 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space equals space 12 over 49 plus 6 over 49 minus 18 over 49 space equals 0
straight b with rightwards arrow on top. straight c with rightwards arrow on top space equals space open parentheses 3 over 7 close parentheses space open parentheses 6 over 7 close parentheses plus open parentheses fraction numerator negative 6 over denominator 7 end fraction close parentheses space open parentheses 2 over 7 close parentheses plus open parentheses 2 over 7 close parentheses space open parentheses fraction numerator negative 3 over denominator 7 end fraction close parentheses space space equals 18 over 49 minus 12 over 49 minus 6 over 49 equals 0

therefore space space space 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 space are space mutually space orthogonal space 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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