38. If $\hat{i},\hat{j}and\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true?

(A) $\hat{i}\times \hat{i}=\overrightarrow{0}$
(B) $\hat{i}\times \hat{k}=\hat{j}$
(C) $\hat{i}·\hat{i}=1$
(D) $\hat{i}·\hat{j}=0$

$\mathrm{<br>}$

Choose the correct answer from the options given below:

The correct answer is (A), (C) and (D) only.

Let us evaluate each of the given statements one by one based on the properties of dot (scalar) and cross (vector) products of orthonormal unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ along the axes OX, OY, and OZ respectively.

1. Analysis of Statement (A):
The cross product of any vector with itself is the zero vector, because the angle $\theta$ between a vector and itself is $0°$, and $\sin (0°)=0$.
Therefore, we have:
$\hat{i}\times \hat{i}=\overrightarrow{0}$
Thus, statement (A) is true.

2. Analysis of Statement (B):
For the unit vectors in a right-handed coordinate system, the cross products follow a cyclic order:
$\hat{i}\times \hat{j}=\hat{k}$,
$\hat{j}\times \hat{k}=\hat{i}$, and
$\hat{k}\times \hat{i}=\hat{j}$.
Since the cross product is anti-commutative, reversing the order introduces a negative sign:
$\hat{i}\times \hat{k}=-(\hat{k}\times \hat{i})=-\hat{j}$
Thus, statement (B) is false (since it claims the result is $\hat{j}$ instead of $-\hat{j}$).

3. Analysis of Statement (C):
The dot product of a unit vector with itself is equal to the square of its magnitude. Since $\hat{i}$ is a unit vector, its magnitude is $1$.
Thus:
$\hat{i}·\hat{i}=|\hat{i}{|}^2=1^2=1$
Thus, statement (C) is true.

4. Analysis of Statement (D):
The dot product of any two mutually perpendicular vectors is $0$ because the angle between them is $90°$, and $\cos (90°)=0$.
Since the axes OX and OY are perpendicular to each other, the unit vectors $\hat{i}$ and $\hat{j}$ are perpendicular.
Thus:
$\hat{i}·\hat{j}=0$
Thus, statement (D) is true.

Comparing our findings, statements (A), (C) and (D) are true, while statement (B) is false.