Unit Vector Properties Explained

This question explores the fundamental properties of unit vectors along the coordinate axes: $\hat{i}$, $\hat{j}$, and $\hat{k}$. Understanding their cross and dot products is essential in vector algebra and physics.

Vector Statement Analysis

Statement (A): $\hat{i} \times \hat{i} = 0$

The cross product of any vector with itself is always the zero vector ($\vec{0}$). This is because the angle between a vector and itself is $0^\circ$, and the sine of $0^\circ$ is $0$. The magnitude of the cross product is given by $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta)$.

Mathematically, for any vector $\vec{A}$, the cross product $\vec{A} \times \vec{A}$ results in the zero vector. Hence, $\hat{i} \times \hat{i} = \vec{0}$.

Therefore, statement (A) is true.

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

The cross product of unit vectors follows a specific cyclic order, often visualized using the right-hand rule for coordinate axes:

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

The cross product is also anti-commutative, meaning $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$.

Following this rule, $\hat{i} \times \hat{k}$ is the reverse of $\hat{k} \times \hat{i}$. Since $\hat{k} \times \hat{i} = \hat{j}$, then $\hat{i} \times \hat{k} = -(\hat{k} \times \hat{i}) = -\hat{j}$.

Therefore, statement (B) is false.

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

The dot product of two vectors is calculated using the formula $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$, where $\theta$ is the angle between them.

Since $\hat{i}$ is a unit vector, its magnitude is $|\hat{i}| = 1$. The angle between a vector and itself is $0^\circ$, and $\cos(0^\circ) = 1$.

So, $\hat{i} \cdot \hat{i} = |\hat{i}| |\hat{i}| \cos(0^\circ) = 1 \times 1 \times 1 = 1$. This property holds true for all unit vectors, meaning $\hat{j} \cdot \hat{j} = 1$ and $\hat{k} \cdot \hat{k} = 1$ as well.

Therefore, statement (C) is true.

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

The unit vectors $\hat{i}$ and $\hat{j}$ are along the x-axis and y-axis, respectively. These axes are perpendicular to each other, so the angle between $\hat{i}$ and $\hat{j}$ is $90^\circ$. For this angle, $\cos(90^\circ) = 0$.

Using the dot product formula:

$\hat{i} \cdot \hat{j} = |\hat{i}| |\hat{j}| \cos(90^\circ) = 1 \times 1 \times 0 = 0$.

This indicates that the dot product of any two distinct unit vectors along the orthogonal coordinate axes is zero. For instance, $\hat{j} \cdot \hat{k} = 0$ and $\hat{k} \cdot \hat{i} = 0$.

Therefore, statement (D) is true.

Summary of True Statements

Based on the detailed analysis of each statement:

• Statement (A) is true.
• Statement (B) is false.
• Statement (C) is true.
• Statement (D) is true.

Consequently, the true statements are (A), (C), and (D).