Question

Find three mutually orthogonal unit vectors in R^3 besides plusminus i, plusminus j, and plusminus k. There are multiple ways to do this and an infinite number of answers. For this problem, we choose a first vector u randomly, choose all but one component of a second vector v randomly, and choose the first component of a third vector w randomly. The other components x, y, and z are chosen so that u, v, and w are mutually orthogonal. Then unit vectors are found based on u, v, and w. Start with u = (1, 1, 2), v = (x, - 1, 2), and w = (1, y, z). The unit vector based on u is (Type exact answers, using radicals as needed.)

Answer 1

Answer:

u=(1,1,2), v=(-3,1,2), w=(1,-2,1/2)

Step-by-step explanation:

Since the vectors must be orthogonal, then they must satisfy that the dot product between them is 0.

Then

$1(x)+1(-1)+2(2)=0\\x=-3$

and $-3(1)-1y+2z=0 \text{ and }\\1+1y+2z=0$.

Solving for y, y=-2z-1 and substituting in the first equation, -3+2z+1+2z=0, then

z=1/2, and y=-2(1/2)-1=-2