Find a unit vector that is orthogonal to both u = (1, 0, 1) v = (0, 1, 1)?

Question

not sure if this apply to this but
U x v = 1
v squared = 2

projv u = 1/2 (0, 1, 1)
= (0, .5, .5 )

u - projvu = (1, 0, 1) - (0, .5, .5)
= 1 , -1/2, 1/2

Answer 1

Let n be the vector normal to both u and v. Take the cross product.

n = u X v = <1, 0, 1> X <0, 1, 1> = <-1, -1, 1>

Now divide by the magnitude of n.

|| n || = √[(-1)² + (-1)² + 1²] = √3

The unit vector orthogonal to u and v is:

n / || n || = <-1/√3, -1/√3, 1/√3>

Answer 2

The vector is (x,y,z) 4x + 5z =0 3x + y = 0 z = (-4/5)x y = -3x with a view to make it a unit vector: x^2 + y^2 + z^2 = a million (a million + sixteen/25 + 9)x^2 = a million x^2 = 25/266 x = 5sqrt(266)/266 And (x,y,z) = 5sqrt(266)/266 (a million, -3, -.8) ~ (.307, -.920, -.245) of path, the alternative unit vector (i.e - (.307, -.920, -.245)) is likewise a valid answer.