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) $\hat{i} \times \hat{i} = \vec{0}$

(B) $\hat{i} \times \hat{k} = \hat{j}$

(C) $\hat{i} \cdot \hat{i} = 1$

(D) $\hat{i} \cdot \hat{j} = 0$

Choose the correct answer from the options given below:

The unit vectors $\hat{i}, \hat{j}, \hat{k}$ point along the X, Y, and Z axes respectively. They are perpendicular to each other and each has magnitude 1.

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For option (A): $\hat{i} \times \hat{i} = \vec{0}$

The cross product of any vector with itself equals the zero vector.

For any vector $\vec{a}$:

$|\vec{a} \times \vec{a}| = |\vec{a}||\vec{a}|\sin(0°)$

$= 0$

Therefore, $\hat{i} \times \hat{i} = \vec{0}$ is true.

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For option (B): $\hat{i} \times \hat{k} = \hat{j}$

The cyclic rule for cross products of unit vectors (in order):

$\hat{i} \times \hat{j} = \hat{k}$

$\hat{j} \times \hat{k} = \hat{i}$

$\hat{k} \times \hat{i} = \hat{j}$

Since cross product is not commutative, going backward in the cycle gives a negative sign:

$\hat{i} \times \hat{k} = -(\hat{k} \times \hat{i})$

$= -\hat{j}$

Therefore, $\hat{i} \times \hat{k} = \hat{j}$ is false.

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For option (C): $\hat{i} \cdot \hat{i} = 1$

The dot product of a vector with itself equals the square of its magnitude:

$\hat{i} \cdot \hat{i} = |\hat{i}|^2$

$= 1^2$

$= 1$

Therefore, $\hat{i} \cdot \hat{i} = 1$ is true.

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For option (D): $\hat{i} \cdot \hat{j} = 0$

The dot product of perpendicular vectors equals zero. Since $\hat{i}$ and $\hat{j}$ are at 90° to each other:

$\hat{i} \cdot \hat{j} = |\hat{i}||\hat{j}|\cos(90°)$

$= 1 \times 1 \times 0$

$= 0$

Therefore, $\hat{i} \cdot \hat{j} = 0$ is true.

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The true statements are (A), (C), and (D).