Find a unit vector that is orthogonal to both i + j and i + k?

Solution:

Let us consider

a = i + j = <1, 1, 0)

b = i + k = <1, 0, 1>

Consider c = + <(cos α, cos β, cos γ)> be the unit vectors which are orthogonal to a and b 

scalar product is c.a = cosα + cosβ = 0

c.b = cosα + cosγ = 0

Which follows c = ± < -cosα, cosα, cosα >

In the respective octant, the directions are equally inclined to the axes so

cosα = ± 1/√3

Therefore, the unit vector is ±(-1/√3, 1/√3, 1/√3).

Find a unit vector that is orthogonal to both i + j and i + k?

Summary:

The unit vector that is orthogonal to both i + j and i + k is ±(-1/√3, 1/√3, 1/√3).