the resultant of A x 0 will be equal to

Options

A. zero
B. A
C. Zero vector
D. unit vector

Answer

To solve the question regarding the resultant of \( \mathbf{A} \times \mathbf{0} \), we can follow these steps:

### Step 1: Understand the Cross Product
The cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the formula:
\[ \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin \theta \, \hat{n} \]
where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of the vectors, \( \theta \) is the angle between the two vectors, and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( \mathbf{A} \) and \( \mathbf{B} \).

### Step 2: Substitute \( \mathbf{B} \) with Zero Vector
In our case, we are calculating \( \mathbf{A} \times \mathbf{0} \). Here, \( \mathbf{0} \) is the zero vector. Thus, we can substitute \( \mathbf{B} \) with \( \mathbf{0} \):
\[ \mathbf{A} \times \mathbf{0} = |\mathbf{A}| |\mathbf{0}| \sin \theta \, \hat{n} \]

### Step 3: Analyze the Magnitude of the Zero Vector
The magnitude of the zero vector \( |\mathbf{0}| \) is 0. Therefore, we can rewrite the equation:
\[ \mathbf{A} \times \mathbf{0} = |\mathbf{A}| \cdot 0 \cdot \sin \theta \, \hat{n} \]

### Step 4: Simplify the Expression
Since any number multiplied by 0 is 0, we have:
\[ \mathbf{A} \times \mathbf{0} = 0 \, \hat{n} \]

### Step 5: Conclusion
The result of the cross product \( \mathbf{A} \times \mathbf{0} \) is the zero vector:
\[ \mathbf{A} \times \mathbf{0} = \mathbf{0} \]

Thus, the answer to the question is that the resultant of \( \mathbf{A} \times \mathbf{0} \) is the zero vector.

### Final Answer
The resultant of \( \mathbf{A} \times \mathbf{0} \) is the zero vector \( \mathbf{0} \).

---