The resultant of two vectors of magnitude 3 units 4 units is 1 unit. What is the value of their dot product. ?

Options

A. `-12` units
B. `-7` units
C. `-1` units
D. zero

Explanation

To find the dot product of two vectors given their magnitudes and the resultant, we can follow these steps:

### Step 1: Understand the Given Information
We have two vectors, \( \vec{A} \) and \( \vec{B} \), with magnitudes:
- \( |\vec{A}| = 3 \) units
- \( |\vec{B}| = 4 \) units
- The resultant \( |\vec{R}| = 1 \) unit

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

### Step 3: Substitute the Values into the Formula
Substituting the given magnitudes and the resultant into the formula:
\[ 1^2 = 3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cos \theta \]
This simplifies to:
\[ 1 = 9 + 16 + 24 \cos \theta \]
\[ 1 = 25 + 24 \cos \theta \]

### Step 4: Solve for \( \cos \theta \)
Rearranging the equation to isolate \( \cos \theta \):
\[ 24 \cos \theta = 1 - 25 \]
\[ 24 \cos \theta = -24 \]
\[ \cos \theta = -1 \]

### Step 5: Determine the Angle \( \theta \)
Since \( \cos \theta = -1 \), this indicates that:
\[ \theta = \pi \text{ radians} \quad (\text{or } 180^\circ) \]
This means the two vectors are in opposite directions.

### Step 6: Calculate the Dot Product
The dot product \( \vec{A} \cdot \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]
Substituting the values:
\[ \vec{A} \cdot \vec{B} = 3 \cdot 4 \cdot (-1) \]
\[ \vec{A} \cdot \vec{B} = -12 \text{ units} \]

### Final Answer
The value of the dot product of the two vectors is \( -12 \) units.

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