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:

Remember: For any unit vector \(\hat{u}\), \(\hat{u} \cdot \hat{u} = 1\) and \(\hat{u} \times \hat{u} = 0\). The dot product gives a scalar, while the cross product gives a vector (zero vector in this case).

Answer 1

The Correct Option is A

We need to evaluate the expression involving dot products and cross product of unit vectors.

Step 1: Recall the properties of unit vectors.
For the standard unit vectors \(\hat{i}, \hat{j}, \hat{k}\):
- \(\hat{i} \cdot \hat{i} = |\hat{i}|^2 = 1^2 = 1\)
- \(\hat{j} \cdot \hat{j} = |\hat{j}|^2 = 1^2 = 1\)
- \(\hat{k} \times \hat{k} = 0\) (cross product of a vector with itself is always the zero vector)

Step 2: Substitute these values into the expression.
\[ \hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k} = 1 - 1 + 0 \]
Step 3: Simplify.
\[ 1 - 1 + 0 = 0 \]
Step 4: Conclusion.
The value of the given expression is 0.
Final Answer: (A) 0