Question

Let v₁ = [-0.5 , v₂ = [0.5 , and v₃ = [-0.5
-0.5 -0.5    0.5

0.5    0.5    0.5

-0.5]    0.5] 0.5]
Find a vector v₄ in R₄ such that the vectors v₁, v₂, v₃, and v₄ are orthonormal.

Answer 1

We presume that v₁ =( -0.5, -0.5,0.5,-0.5)^T = (-1/2,-1/2,1/2,-1/2)^T , v₂ =( 0.5,- 0.5,0.5,0.5)^T = (1/2,-1/2,1/2,1/2)T and v₃ =( -0.5,0.5,0.5,0.5)^T = (-1/2,1/2,1/2,1/2)^T. Then v₁,v₂,v₃ are orthonormal vectors.

Let v₄ = (x,y,z,w)^T be a vector in R⁴ such that the vectors v₁, v₂, v₃, and v₄ are orthonormal.

Then v₄.v₁ = 0 or, (x,y,z,w)^T. (-1/2,-1/2,1/2,-1/2)^T = 0 or, -(1/2)x-(1/2)y+(1/2)z+(1/2) w = 0 or, x+y-z+w = 0…(1)

Also, v₄.v₂ = 0 or, (x,y,z,w)^T. (1/2,-1/2,1/2,1/2)^T = 0 or, (1/2)x-(1/2)y+(1/2)z+(1/2) w = 0 or, x-y+z+w = 0…(2) and

v₄.v₃ = 0 or, (x,y,z,w)^T. (-1/2,1/2,1/2,1/2)^T = 0 or, -(1/2)x+(1/2)y+(1/2)z+(1/2) w = 0 or, x-y-z-w = 0…(3)

The coefficient matrix of this linear system is A(say) =

1	1	-1	-1
1	-1	1	1
1	-1	-1	-1

To solve the linear system , we will reduce A to its RREF which is

1	0	0	1
0	1	0	1
0	0	1	1

This implies that x+w = 0 or, x = -w, y +w= 0 or, y = -w and z+w = 0 or, z = - w.

Also, since v₄ is a unit vector, we have (x²+y²+z²+w²) = 1 or, 4w² = 1 or, w = 1/2 so that v₄ = (-1/2,-1/2,-1/2, 1/2)^T = (-0.5,-0.5,-0.5,0.5)^T.