If $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then

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

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

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

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

Choose the correct answer from the options given below:

The unit vectors $\hat{i}$, $\hat{j}$, $\hat{k}$ have length 1 and lie along the X, Y, and Z axes respectively.

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For statement (A): $\hat{i} \times \hat{j} = \hat{k}$

The cross product follows cyclic order: $\hat{i} → \hat{j} → \hat{k} → \hat{i}$

Going forward in the cycle gives a positive result.

$\hat{i} \times \hat{j}$ goes forward from $\hat{i}$ to $\hat{j}$, giving $+\hat{k}$

Statement (A) is TRUE.

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

Using the cyclic order:

$\hat{k} \times \hat{i}$ goes forward in the cycle from $\hat{k}$ to $\hat{i}$, giving $+\hat{j}$

The statement claims $-\hat{j}$.

Statement (B) is FALSE.

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For statement (C): $\hat{j} \cdot \hat{j} = 1$

The dot product of any unit vector with itself equals 1.

$\vec{a} \cdot \vec{a} = |\vec{a}|^2$

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

$\hat{j} \cdot \hat{j} = 1^2$

$\hat{j} \cdot \hat{j} = 1$

Statement (C) is TRUE.

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For statement (D): $\hat{j} \cdot \hat{k} = 0$

The dot product of perpendicular vectors equals zero.

$\hat{j}$ (Y-axis) and $\hat{k}$ (Z-axis) are perpendicular.

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

$\hat{j} \cdot \hat{k} = 1 \times 1 \times 0$

$\hat{j} \cdot \hat{k} = 0$

Statement (D) is TRUE.

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TRUE statements: (A), (C), and (D)

FALSE statement: (B)

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