Starting from the first orthogonal vector u₁=(3,-2,4), follow these steps:
Calculate the projection of v₂=(4,2,1) onto u₁:
proj(v₂) = [(v₂ ⋅ u₁)/(u₁ ⋅ u₁)] × u₁ = 12/29 × (4,2,1) = (1.24,-0.83,1.66)
Subtract the projection from v₂:
v₂ - proj(v₂) = (4,2,1) - (1.24,-0.83,1.66) = (2.76, 2.83, -0.66)
The result is the vector u₂.
To confirm their orthogonality, you can compute the dot product and verify that it is equal to zero.
Read more