SOLUTION: Please help. Thank you. Find all values of x and y so that u=xi+2yj-8k is perpendicular to both v=2i-j+k and w=3i+2j-4k.

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 = .


Then from eq(1')  2y = 2x-8 =  =  = .


Answer.  x= ,  y= .