The dot product of two vectors of magnitudes 3 units and 5 units cannot be :-
(i) -20 (ii) 16 (iii) -10 (br) 14

Options

A. (i,iii)
B. (i,ii)
C. (i, iv)
D. (ii, iii, iv)

Answer

To solve the problem of determining which of the given values cannot be the dot product of two vectors with magnitudes 3 units and 5 units, we can follow these steps:

### Step 1: Understand the Dot Product Formula
The dot product of two vectors **A** and **B** can be expressed as:
\[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \]
where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of the vectors and \( \theta \) is the angle between them.

### Step 2: Identify the Magnitudes
From the problem, we have:
- Magnitude of vector A, \( |\mathbf{A}| = 3 \) units
- Magnitude of vector B, \( |\mathbf{B}| = 5 \) units

### Step 3: Calculate the Maximum and Minimum Values of the Dot Product
The maximum value of \( \cos \theta \) is 1 (when \( \theta = 0^\circ \)), and the minimum value is -1 (when \( \theta = 180^\circ \)). Thus, we can calculate the extreme values of the dot product:
- Maximum dot product:
\[ \mathbf{A} \cdot \mathbf{B}_{\text{max}} = 3 \times 5 \times 1 = 15 \]
- Minimum dot product:
\[ \mathbf{A} \cdot \mathbf{B}_{\text{min}} = 3 \times 5 \times (-1) = -15 \]

### Step 4: Determine the Range of the Dot Product
From the calculations, we find that the dot product can vary between:
\[ -15 \leq \mathbf{A} \cdot \mathbf{B} \leq 15 \]

### Step 5: Analyze the Given Options
Now we will check each of the provided options to see if they fall within the range of -15 to 15:
1. **-20**: This value is less than -15, so it **cannot** be the dot product.
2. **16**: This value is greater than 15, so it **cannot** be the dot product.
3. **-10**: This value is within the range of -15 to 15, so it **can** be the dot product.
4. **14**: This value is also within the range of -15 to 15, so it **can** be the dot product.

### Conclusion
The values that cannot be the dot product of the two vectors are **-20** and **16**. Therefore, the correct answer to the question is:
- **The dot product cannot be -20 or 16.**