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

Answer

To solve the problem, we need to find the dot product of two vectors given their magnitudes and the magnitude of their resultant. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We have two vectors:
- Vector A with a magnitude of \( |A| = 3 \) units
- Vector B with a magnitude of \( |B| = 4 \) units
- The magnitude of their resultant vector \( |R| = 1 \) unit

### Step 2: Use the Formula for the Resultant of Two Vectors
The magnitude of the resultant \( R \) of two vectors can be expressed using the formula:
\[ |R| = \sqrt{|A|^2 + |B|^2 + 2|A||B|\cos(\theta)} \]
where \( \theta \) is the angle between the two vectors.

### Step 3: Substitute the Known Values
Substituting the known values into the formula:
\[ 1 = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos(\theta)} \]
This simplifies to:
\[ 1 = \sqrt{9 + 16 + 24\cos(\theta)} \]
\[ 1 = \sqrt{25 + 24\cos(\theta)} \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ 1^2 = (25 + 24\cos(\theta)) \]
\[ 1 = 25 + 24\cos(\theta) \]

### Step 5: Solve for \( \cos(\theta) \)
Rearranging the equation gives:
\[ 24\cos(\theta) = 1 - 25 \]
\[ 24\cos(\theta) = -24 \]
\[ \cos(\theta) = -1 \]

### Step 6: Determine the Angle \( \theta \)
Since \( \cos(\theta) = -1 \), this implies:
\[ \theta = 180^\circ \]
This means the two vectors are in opposite directions.

### Step 7: Calculate the Dot Product
The dot product of two vectors \( A \) and \( B \) is given by:
\[ A \cdot B = |A| |B| \cos(\theta) \]
Substituting the values we have:
\[ A \cdot B = 3 \cdot 4 \cdot (-1) \]
\[ A \cdot B = -12 \]

### Final Answer
The value of their dot product is \( -12 \).

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