Question

Prove that

i. [i  j  k] = [j  k  i] = [k  i  j] = 1

ii. [i  k  j] = [k  j  i] = [j  i  k] = -1

Answer 1

i. [i  j  k] = [j  k  i] = [k  i  j] = 1

Let, i, j, k be unit vectors in the direction of positive X -axis, Y -axis, Z -axis respectively.

Hence,

Formulae :

a) Dot Products :

b) Cross Products :

c) Scalar Triple Product :

From eq(1), eq(2) and eq(3),

[i  j  k] = [j  k  i] = [k  i  j] = 1

Hence Proved.

Notes :

1. A cyclic change of vectors in a scalar triple product does not change its value i.e.

2. Scalar triple product of unit vectors taken in a clockwise direction is 1, and that of unit vectors taken in anticlockwise direction is -1

Let, i, j, k be unit vectors in the direction of positive X -axis, Y -axis, Z -axis respectively.

Hence,

Formulae :

a) Dot Products :

b) Cross Products :

c) Scalar Triple Product :

Answer :

From eq(1), eq(2) and eq(3),

Hence Proved.

Notes :

1. A cyclic change of vectors in a scalar triple product does not change its value i.e.

2. Scalar triple product of unit vectors taken in a clockwise direction is 1, and that of unit vectors taken in anticlockwise direction is -1