Determine the value of c so that for the real x, vectors cx `hati - 6 hatj - 3 hatk and xhati + 2hatj + 2cx hatk` make an obtuse angle with each other .

Explanation

To determine the value of \( c \) such that the vectors \( \mathbf{A} = cx \hat{i} - 6 \hat{j} - 3 \hat{k} \) and \( \mathbf{B} = x \hat{i} + 2 \hat{j} + 2cx \hat{k} \) make an obtuse angle with each other, we will use the property of dot product. The angle between two vectors is obtuse if their dot product is less than zero.

### Step 1: Write the dot product of the vectors

The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:

\[ \mathbf{A} \cdot \mathbf{B} = (cx)(x) + (-6)(2) + (-3)(2cx) \]

### Step 2: Simplify the dot product expression

Calculating the dot product:

\[ \mathbf{A} \cdot \mathbf{B} = cx^2 - 12 - 6cx \]

### Step 3: Set up the inequality for obtuse angle

For the vectors to form an obtuse angle, we require:

\[ cx^2 - 6cx - 12 < 0 \]

### Step 4: Factor the quadratic expression

We can rewrite the inequality as:

\[ -cx^2 + 6cx + 12 > 0 \]

### Step 5: Identify the coefficients for the quadratic inequality

The quadratic can be expressed as:

\[ -cx^2 + 6cx + 12 \]

Here, \( a = -c \), \( b = 6c \), and \( c = 12 \).

### Step 6: Find the discriminant

The discriminant \( D \) of the quadratic equation is given by:

\[ D = b^2 - 4ac = (6c)^2 - 4(-c)(12) = 36c^2 + 48c \]

### Step 7: Set the discriminant greater than or equal to zero

For the quadratic to have real roots, we need:

\[ 36c^2 + 48c \geq 0 \]

### Step 8: Factor the discriminant expression

Factoring gives:

\[ 12c(3c + 4) \geq 0 \]

### Step 9: Solve the inequality

Setting each factor to zero gives:

1. \( 12c = 0 \) → \( c = 0 \)
2. \( 3c + 4 = 0 \) → \( c = -\frac{4}{3} \)

### Step 10: Determine the intervals

The critical points divide the number line into intervals. We test the intervals:

- For \( c < -\frac{4}{3} \): Choose \( c = -2 \) → \( 12(-2)(3(-2) + 4) = 12(-2)(-6 + 4) < 0 \)
- For \( -\frac{4}{3} < c < 0 \): Choose \( c = -1 \) → \( 12(-1)(3(-1) + 4) = 12(-1)(-3 + 4) > 0 \)
- For \( c > 0 \): Choose \( c = 1 \) → \( 12(1)(3(1) + 4) > 0 \)

### Step 11: Conclusion

The solution to the inequality \( 12c(3c + 4) \geq 0 \) gives us:

\[ c \in \left[-\frac{4}{3}, 0\right] \]

Thus, the values of \( c \) for which the vectors make an obtuse angle are:

\[ c \leq 0 \quad \text{and} \quad c \geq -\frac{4}{3} \]