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

The given mathematical expression is:

\(\hat{i} \cdot \hat{i} - \hat{j} \cdot \hat{j} + \hat{k} \times \hat{k}\)

Let's break it down term by term:

1. Dot Product:

   The dot product of a unit vector with itself is always 1. Therefore,

   • \(\hat{i} \cdot \hat{i} = 1\)
   • \(\hat{j} \cdot \hat{j} = 1\)
2. Cross Product:

   The cross product of any vector with itself is always 0 because the angle between the same vectors is 0 degrees, and the sine of 0 is 0. Thus,

   • \(\hat{k} \times \hat{k} = 0\)

Substituting these values back into the expression gives:

1 - 1 + 0 = 0

Therefore, the value of the expression is 0.

Conclusion: The correct answer is 0.