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If u = i + 3j - 2k, and v = 4i - 2j - 4k.

0 votes

Find ( 2u + v ) ( u - 2v ).

asked Jun 8, 2013 in PRECALCULUS by anonymous Apprentice

2 Answers

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Given  equations are u = i + 3 j-2k , and  v =  4i - 2j  -4k

u = i+3j -2k      ------->  (1)

v = 4i -2j -4k    ------->  (2)

Subsutitute  (1) and (2)  valus  in  the  equation  (2u+v)  (u-2v)

    ----> 2u+v  = 2 ( i +3j - 2k ) + (4i - 2j- 4 k )

                       = ( 2i + 6j  - 4k )  + (4i - 2j - 4k )

                       = 2i +  6j  -4k  +  4i  - 2j  - 4k            (Addtion  property  )

                       = 6i  + 4j  - 8k                    ->  (3)

   -------> (u -2v )  =  (  i + 3j  -2k  )  -  2  (  4i -  2j  - 4k  )

                             =  i  +  3j  - 2k  - 8 i  + 4j  + 8k                

                             =  - 7i  + 7j  +6k           ->   (4 )

                Multiply  (3 ) &  (4 )

----->  ( 2u + v)  (u - 2 v )  =   ( 6i + 4j  - 8k )  (  7i  + 7j + 6k )

                                              =  6i ( 7i + 7j + 6 k )  +  4j  (  7i +  7j  + 6k )  -  8k (  7i  +  7j  + 6k )

                                              =[ 6i (7i ) +  6i ( 7j )  + 6i (6k ]  +  [  4j ( 7i )  +4j ( 7j )  +  4 j ( 6k ) ]  -  [ 8k ( 7i )  + 8k ( 7j )  +8k(6k) ]                                            

                                             = [ 42 ( i X i )  + 42 ( i X j )  + 36 ( i X k ) ]  + [ 28 ( j x i ) + 28 ( j X j ) + 24 (  j X K ) ]  - [ 56 (K X i )  + 56 ( k x j ) +  48 ( k x k ) ]

                                           = [ 42 ( 0 ) + 42 ( k  )  +  36 ( j ) ]  +  [ 28  ( k ) +  28  ( 0 )  + 24 ( i ) ]  - [  56 ( j )  + 56 (  i )  + 48  ( 0) ]

        [ Product  of   i x  i = 0 ,   j x j = 0 ,   k x k = 0 ,  i x j  =k ,   j x k = i and  k x i = j ]

                                          = [ 0 + 42 k  + 36 j ]  +  [ 28 k  +  0  +  24 i  ]  - [ 56 j  + 56  i  + 0 ]

                                          = [  42 k +  36 j  ]  +  [ 28 k  + 24 i  ]  - [ 56 j  +  56  i ]

                                          = [  42 k + 36 j  + 28 k + 24 i  - 56 j  - 56  i  ]

                                          = [  -32 i  - 20 j  + 90 k  ]

    ( 2u +  v )  (  u  -  2v )   =  -  32 i  - 20 j  +  90  k

The solution is -  32 i  - 20 j  +  90  k.

answered Jun 8, 2013 by goushi Pupil
0 votes

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The value of (2u + v) · (u - 2v) = - 62 and (2u + v) × (u - 2v) = 80i + 20j + 70k.

answered Jul 16, 2014 by casacop Expert

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