Which of the following sets of vectors in R3 is orthonormal

Which of the following sets of vectors in R^3 is orthonormal? (The inner product used is the dot product.) {[0 1 0], [Squareroot 2/2 0 Squareroot 2/2], [-Squareroot 2/2 0 Squareroot 2/2]} {[1 0 0], [Squareroot 2/2 0 Squareroot 2/2], [-Squareroot 2/2 0 Squareroot 2/2]} {[0 0 1], [0 1 1], [1 -1 0]} {[0 1 0], [0 Squareroot 2/2 Squareroot 2/2], [-Squareroot 2/2 0 Squareroot 2/2]} {[0 0 1], [0 1 0], [1 1 0]}

Solution

Orthonormal vectors are those vectors which are perpendicular to each other & of unit length.

Now take the first case

Dot product of 1st & 2nd=0,2nd& 3rd=-1/2+1/2=0,3rd & 1st=0

Length of vectors are 1,1,1

So first case is orthonormal.

Take 2 and case

Dot product of 1st & 2nd=1/2#0

So it is not orthonormal

Take 3rd case

Dot product of 1st &2nd vectors=1

So it is not orthonormal

Take 4 th case

Dot product of 2nd & 3rd case is 1/2

So it is not orthonormal

Take 5th case

Dot product of 2nd &3rd vectors is 1#0

So it is not orthonormal.

So only first set is orthonormal.