Perpendicular Vectors. I need help please

Question

Find two vectors u and v such that: (i) Both the vector u and v are perpendicular to the vector w =(1,0,-1) and (ii) the vector u is perpendicular to the vector v.

Answer 1

For vectors to be perpendicular, their dot products must be zero.

i)

u•w = u_x-u_z = 0

v•w = v_x-v_z =0

ii)

u•v = u_xv_x+u_yv_y+u_zv_z = 0

2u_zv_z + u_yv_y = 0 --> 2u_zv_z= -u_yv_y

Let u_z=1, v_z=1, u_y=-2, v_y=1

u_xv_x -2 +1 = 0 --> u_xv_x=1

Let u_x= v_x= 1

u = (1,-2,1) v = (1,1, 1)

Checking:

u • v = 1-2+1 = 0

u • w = 1+0-1 = 0

v • w = 1 +0 -1 = 0