What will be the value of cross product of two vectors of magnitudes 4 and 5, if their resultant is 1?

Explanation

To solve the problem of finding the cross product of two vectors with magnitudes 4 and 5, given that their resultant is 1, we can follow these steps:

### Step 1: Understand the Given Information
We have two vectors, \( \vec{A} \) and \( \vec{B} \), with magnitudes:
- \( |\vec{A}| = 4 \)
- \( |\vec{B}| = 5 \)

The resultant of these two vectors is given as:
- \( |\vec{R}| = 1 \)

### Step 2: Use the Resultant Formula
The formula for the magnitude of the resultant \( |\vec{R}| \) of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ |\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2 |\vec{A}| |\vec{B}| \cos \theta} \]
where \( \theta \) is the angle between the two vectors.

### Step 3: Substitute the Known Values
Substituting the known values into the formula:
\[ 1 = \sqrt{4^2 + 5^2 + 2 \cdot 4 \cdot 5 \cdot \cos \theta} \]
Calculating \( 4^2 \) and \( 5^2 \):
\[ 1 = \sqrt{16 + 25 + 40 \cos \theta} \]
This simplifies to:
\[ 1 = \sqrt{41 + 40 \cos \theta} \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ 1^2 = 41 + 40 \cos \theta \]
This gives us:
\[ 1 = 41 + 40 \cos \theta \]

### Step 5: Solve for \( \cos \theta \)
Rearranging the equation:
\[ 40 \cos \theta = 1 - 41 \]
\[ 40 \cos \theta = -40 \]
Dividing both sides by 40:
\[ \cos \theta = -1 \]

### Step 6: Determine the Angle \( \theta \)
The value \( \cos \theta = -1 \) corresponds to:
\[ \theta = 180^\circ \]
This indicates that the two vectors are in opposite directions.

### Step 7: Calculate the Cross Product
The formula for the cross product \( \vec{A} \times \vec{B} \) is given by:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]
Substituting the known values:
\[ |\vec{A} \times \vec{B}| = 4 \cdot 5 \cdot \sin(180^\circ) \]
Since \( \sin(180^\circ) = 0 \):
\[ |\vec{A} \times \vec{B}| = 4 \cdot 5 \cdot 0 = 0 \]

### Final Answer
The value of the cross product of the two vectors is:
\[ \boxed{0} \]