The value of c so that for all real x, the vectors `cxhatk + 3hatj, xhati + 2hatj + 2cx hatk` make an obtuse angle are:

Options

A. `c lt 0`
B. `0 lt c lt 4//3`
C. `-4//3 lt c lt 0`
D. `c gt 0`

Answer

To solve the problem, we need to find the value of \( c \) such that the vectors \( \mathbf{A} = c\hat{k} + 3\hat{j} \) and \( \mathbf{B} = x\hat{i} + 2\hat{j} + 2cx\hat{k} \) make an obtuse angle for all real \( x \).

### Step-by-Step Solution:

1. **Understanding the Condition for Obtuse Angle:**
The angle between two vectors is obtuse if their dot product is negative. Therefore, we need to find the dot product \( \mathbf{A} \cdot \mathbf{B} \) and set it to be less than zero.

2. **Calculate the Dot Product:**
The vectors are:
\[ \mathbf{A} = 0\hat{i} + 3\hat{j} + c\hat{k} \]
\[ \mathbf{B} = x\hat{i} + 2\hat{j} + 2cx\hat{k} \]
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (0)(x) + (3)(2) + (c)(2cx) = 6 + 2c^2x \]

3. **Set the Dot Product Less Than Zero:**
For the angle to be obtuse, we need:
\[ 6 + 2c^2x < 0 \]
Rearranging gives:
\[ 2c^2x < -6 \]
\[ c^2x < -3 \]

4. **Analyzing the Inequality:**
Since \( x \) can take any real value, for the inequality \( c^2x < -3 \) to hold for all \( x \), \( c^2 \) must be positive and \( x \) must be negative. However, since \( x \) can also be positive, we need to analyze the discriminant of the quadratic formed by rearranging the inequality.

5. **Forming a Quadratic Equation:**
The expression \( 2c^2x + 6 < 0 \) can be treated as a quadratic in \( x \):
\[ 2c^2x + 6 = 0 \]
The discriminant \( D \) of this quadratic must be less than zero for there to be no real solutions:
\[ D = b^2 - 4ac = 0^2 - 4(2c^2)(6) < 0 \]
\[ -48c^2 < 0 \]

6. **Conclusion from the Discriminant:**
Since \( -48c^2 < 0 \) implies \( c^2 > 0 \), we conclude that \( c \) must be non-zero. Thus:
\[ c \neq 0 \]

7. **Final Condition:**
The only requirement for \( c \) is that it must be greater than zero:
\[ c > 0 \]

### Final Answer:
The value of \( c \) such that the vectors make an obtuse angle for all real \( x \) is \( c > 0 \).