Can the resultant of two vector be zero

Options

A. Yes, when the 2 vectors are same in magnitude and direction
B. No
C. Yes, when the 2 vectors are same in magnitude but opposite in sense
D. Yes, when the 2 vectors are same in magnitude making an angle of `(2pi)/(3)` with each other

Answer

To determine if the resultant of two vectors can be zero, we can analyze the situation mathematically. Here’s a step-by-step solution:

### Step 1: Understanding Vectors
Vectors have both magnitude and direction. The resultant of two vectors is the vector sum of those two vectors.

### Step 2: Resultant of Two Vectors
The resultant \( R \) of two vectors \( \vec{A} \) and \( \vec{B} \) can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
where \( A \) and \( B \) are the magnitudes of the vectors, and \( \theta \) is the angle between them.

### Step 3: Setting the Resultant to Zero
To find out if the resultant can be zero, we set \( R = 0 \):
\[ 0 = A^2 + B^2 + 2AB \cos \theta \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ A^2 + B^2 + 2AB \cos \theta = 0 \]

### Step 5: Analyzing the Condition
For the above equation to hold true, the left-hand side must equal zero. This can happen if:
- The magnitudes of the vectors are equal, i.e., \( A = B \).
- The angle \( \theta \) between them is \( 180^\circ \) (which means they are in opposite directions).

### Step 6: Substituting Values
If we substitute \( A = B \):
\[ A^2 + A^2 + 2A^2 \cos(180^\circ) = 0 \]
Since \( \cos(180^\circ) = -1 \), we have:
\[ 2A^2 - 2A^2 = 0 \]
This confirms that the resultant is indeed zero.

### Conclusion
Thus, the resultant of two vectors can be zero if they are equal in magnitude and act in opposite directions.