Question

The value of expression \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} \) is

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

In vector problems on multiple-choice tests, if an expression seems mathematically invalid (like adding scalars and vectors), check for a likely typo. Often, an operator is incorrect. Look for patterns in the expression to deduce the intended question. Here, the pattern of self-dot-products is the strongest clue.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This expression involves the dot product (\( \cdot \)) and cross product (\( \times \)) of the standard orthogonal unit vectors \( \hat{i}, \hat{j}, \hat{k} \). A key issue is that the expression as written attempts to add scalars (\( \hat{i} \cdot \hat{i} \) and \( \hat{j} \cdot \hat{j} \)) and a vector (\( \hat{k} \times \hat{k} \)), which is mathematically undefined. This strongly implies a typographical error in the original question paper.
Step 2: Key Formula or Approach:
The standard properties of unit vectors are:
Dot product of a unit vector with itself: \( \hat{i} \cdot \hat{i} = 1 \), \( \hat{j} \cdot \hat{j} = 1 \), \( \hat{k} \cdot \hat{k} = 1 \).
Cross product of any vector with itself is the zero vector: \( \vec{A} \times \vec{A} = \vec{0} \). Therefore, \( \hat{k} \times \hat{k} = \vec{0} \).
Given the invalidity of adding a scalar to a vector, we must assume the most likely typo. The pattern of the first two terms suggests the third term was also intended to be a dot product. Thus, we will solve the problem by assuming the expression is \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \).
Step 3: Detailed Calculation:
Assuming the corrected expression is \( \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \cdot \hat{k} \).
Let's evaluate each term based on the properties above:
\( \hat{i} \cdot \hat{i} = |\hat{i}|^2 = 1^2 = 1 \)
\( \hat{j} \cdot \hat{j} = |\hat{j}|^2 = 1^2 = 1 \)
\( \hat{k} \cdot \hat{k} = |\hat{k}|^2 = 1^2 = 1 \)
Now, substitute these values back into the corrected expression: \[ \text{Value} = (\hat{i} \cdot \hat{i}) - (\hat{j} \cdot \hat{j}) + (\hat{k} \cdot \hat{k}) \] \[ \text{Value} = 1 - 1 + 1 \] \[ \text{Value} = 1 \] Step 4: Final Answer:
The calculated value is 1.