Question

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) $\hat{i} . \hat{i} = 1$
(D) $\hat{i} . \hat{j} = 0$
Choose the correct answer from the options given below:

• A. (A) and (B) only
• B. (A), (C) and (D) only
• C. (A) only
• D. (A), (B), (C) and (D)

Hint:

Remember the geometric interpretations: the dot product measures projection and is related to the cosine of the angle, while the cross product measures the area of the parallelogram formed by the vectors and is related to the sine of the angle, yielding a vector perpendicular to both.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests the fundamental properties of the standard basis vectors ($\hat{i}, \hat{j}, \hat{k}$) in a 3D Cartesian coordinate system, specifically their dot products and cross products.
Step 3: Detailed Explanation:
Let's evaluate each statement based on the definitions of dot and cross products.
(A) $\hat{i \times \hat{i} = \vec{0}$}
The cross product of any vector with itself is the zero vector ($\vec{0}$). This is because the angle between the vectors is 0, and $\sin(0) = 0$. The magnitude is $|\hat{i}||\hat{i}|\sin(0) = 1 \cdot 1 \cdot 0 = 0$. So, statement (A) is true.
(B) $\hat{i \times \hat{k} = \hat{j}$}
The cross products of the basis vectors follow a cyclic rule for a right-handed coordinate system: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, and $\hat{k} \times \hat{i} = \hat{j}$. Reversing the order of a cross product introduces a negative sign.
Therefore, $\hat{i} \times \hat{k} = -(\hat{k} \times \hat{i}) = -\hat{j}$. So, statement (B) is false.
(C) $\hat{i \cdot \hat{i} = 1$}
The dot product of a vector with itself is the square of its magnitude. Since $\hat{i}$ is a unit vector, its magnitude is 1. Thus, $\hat{i} \cdot \hat{i} = |\hat{i}|^2 = 1^2 = 1$. So, statement (C) is 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 $\cos(90^\circ) = 0$. So, statement (D) is true.
Step 4: Final Answer:
The true statements are (A), (C), and (D). This corresponds to option (2).