Question 1114538
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The condition that the vector  u = xi+2yj-8k  is perpendicular to the vector  v = 2i-j+k  is that their scalar product is equal to zero:


    x*2 - 2*y - 8 = 0.     (1)


The condition that the vector  u = xi+2yj-8k  is perpendicular to the vector  w = 3i+2j-4k  is that their scalar product is equal to zero:


    x*3 + 4y + 32 = 0.     (2)


Thus you have this system of 2 equations in 2 unknowns 

    2x - 2y =   8          (1')
    3x + 4y = -32          (2')


Apply the Elimination method.  For it, multiply eq(1') by 2.  Keep eq(2') as is:


    4x - 4y =  16          (1'')
    3x + 4y = -32          (2'')


Now add equations (1'') and (2'')


    7x = 16 - 32  ====>  x = {{{-16/7}}}.


Then from eq(1')  2y = 2x-8 = {{{2*(-16/7)+8}}} = {{{-32/7+8}}} = {{{24/7}}}.


<U>Answer</U>.  x= {{{-16/7}}},  y= {{{24/7}}}.
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