To show that,, is a right handed orthogonal system of unit vectors, we need to prove the following –

(a)

(b)

(c)

(d)

Let us consider each of these one at a time.

(a) Recall the magnitude of the vector is

First, we will find.

Now, we will find.

Finally, we will find.

Hence, we have

(b) Now, we will evaluate the vector

Recall the cross product of two vectors and is

Taking the scalar common, here, we have (a₁, a₂, a₃) = (2, 3, 6) and (b₁, b₂, b₃) = (3, –6, 2)

Hence, we have.

(c) Now, we will evaluate the vector

Taking the scalar common, here, we have (a₁, a₂, a₃) = (3, –6, 2) and (b₁, b₂, b₃) = (6, 2, –3)

Hence, we have.

(d) Now, we will evaluate the vector

Taking the scalar common, here, we have (a₁, a₂, a₃) = (6, 2, –3) and (b₁, b₂, b₃) = (2, 3, 6)

Hence, we have.

Thus,,, is also another right handed orthogonal system of unit vectors.