Step 1: Conceptual Foundation:
This question assesses comprehension of the fundamental properties of standard basis vectors ($\hat{i}, \hat{j}, \hat{k}$) in a 3D Cartesian system, specifically their dot and cross product behaviors.
Step 3: Detailed Analysis:
Each statement will be evaluated using the definitions of dot and cross products.
(A) $\hat{i} \times \hat{i} = \vec{0}$
The cross product of any vector with itself yields the zero vector ($\vec{0}$). This is due to the angle between the vectors being 0, for which $\sin(0) = 0$. The magnitude is $|\hat{i}||\hat{i}|\sin(0) = 1 \cdot 1 \cdot 0 = 0$. Therefore, statement (A) is true.
(B) $\hat{i} \times \hat{k} = \hat{j}$
In a right-handed coordinate system, the cross products of basis vectors follow a cyclic pattern: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, and $\hat{k} \times \hat{i} = \hat{j}$. Reversing the order of a cross product negates the result.
Consequently, $\hat{i} \times \hat{k} = -(\hat{k} \times \hat{i}) = -\hat{j}$. Thus, statement (B) is false.
(C) $\hat{i} \cdot \hat{i} = 1$
The dot product of a vector with itself equals the square of its magnitude. As $\hat{i}$ is a unit vector, its magnitude is 1. Hence, $\hat{i} \cdot \hat{i} = |\hat{i}|^2 = 1^2 = 1$. Statement (C) is therefore true.
(D) $\hat{i} \cdot \hat{j} = 0$
The vectors $\hat{i}$ and $\hat{j}$ are orthogonal (perpendicular), meaning the angle between them is 90 degrees. The dot product of orthogonal vectors is zero, as $\cos(90^\circ) = 0$. Statement (D) is true.
Step 4: Conclusion:
The true statements are (A), (C), and (D). This corresponds to option (2).