Let  Find a vector  which is perpendicular to both  from M

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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.

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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.


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 space straight b with rightwards arrow on top space equals space straight i with hat on top space minus space 4 straight j with hat on top space plus 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 space plus space straight j with hat on top space minus space straight k with hat on top
Let  straight d with rightwards arrow on top space equals space straight d subscript 1 straight i with hat on top plus straight d subscript 2 straight j with hat on top plus straight d subscript 3 straight k with hat on top
Since straight d with rightwards arrow on top is perpendicular to straight a with rightwards arrow on top space and space straight b with rightwards arrow on top
therefore space space straight d with rightwards arrow on top space. space straight a with rightwards arrow on top space equals space 0 space space and space straight d with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0
therefore space space space space 4 space straight d subscript 1 space plus space 5 space straight d subscript 2 space minus space straight d subscript 3 space equals space 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
and   straight d subscript 1 minus 4 straight d subscript 2 plus 5 straight d subscript 3 space equals 0                                                                     ...(2)
Again,                  straight d with rightwards arrow on top. space straight i with hat on top space equals space 21
rightwards double arrow space space space 3 space straight d subscript 1 space plus space straight d subscript 2 space minus straight d subscript 3 space equals space 21                                                                 ...(3)
Subtracting (3) from (1), we get, 
                          straight d subscript 1 plus 4 space straight d subscript 2 space equals space minus 21                                                       ...(4)               
Multiply (2) by 1 and (3) by 5, we get.
    straight d subscript 1 minus 4 space straight d subscript 2 plus 5 space straight d subscript 3 space equals 0
   15 space straight d subscript 1 space plus space 5 space straight d subscript 2 space minus space 5 space straight d subscript 3 space equals space 105
Adding,      16 space straight d subscript 1 space plus space straight d subscript 2 space equals space 105                                                           ...(5)
therefore space space space space 64 space straight d subscript 1 space plus space 4 space straight d subscript 2 space equals space 420                                                                      
Subtracting it from (4), we get,
                    negative 63 space straight d subscript 1 space equals space minus 441 space space space space space space space space space space space rightwards double arrow space space space space straight d subscript 1 space equals space 7
From (5),      112 plus straight d subscript 2 space equals space 105 space space space space space space space rightwards double arrow space space space space space space straight d subscript 2 space equals space minus 7
From (1), 28 minus 35 space minus straight d subscript 3 space equals 0 space space space space space space space space space space space space space space space space rightwards double arrow space space space straight d subscript 3 space equals space minus 7
therefore space space space space straight d with rightwards arrow on top space equals 7 straight i with hat on top space minus space 7 straight j with hat on top space minus space 7 straight k with hat on top
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