The vectors `(2hati - mhatj+ 3mk)` and `{(1 + m) hati -2m hatj + hatk}` include and acute angle for

Options

A. all values of m
B. `m lt -2 or m gt - 1//2`
C. m = -1/2
D. `m in [ -2, - 1/2]`

Answer

To determine the values of \( m \) for which the vectors \( \mathbf{A} = 2\hat{i} - m\hat{j} + 3m\hat{k} \) and \( \mathbf{B} = (1 + m)\hat{i} - 2m\hat{j} + \hat{k} \) include an acute angle, we will follow these steps:

### Step 1: Write down the vectors
We have:
\[ \mathbf{A} = 2\hat{i} - m\hat{j} + 3m\hat{k} \]
\[ \mathbf{B} = (1 + m)\hat{i} - 2m\hat{j} + \hat{k} \]

### Step 2: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \)
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(1 + m) + (-m)(-2m) + (3m)(1) \]
Calculating this:
\[ \mathbf{A} \cdot \mathbf{B} = 2(1 + m) + 2m^2 + 3m \]
\[ = 2 + 2m + 2m^2 + 3m \]
\[ = 2m^2 + 5m + 2 \]

### Step 3: Set the dot product greater than zero
For the vectors to include an acute angle, the dot product must be positive:
\[ 2m^2 + 5m + 2 > 0 \]

### Step 4: Factor the quadratic expression
We can factor the quadratic:
\[ 2m^2 + 5m + 2 = (m + 2)(2m + 1) \]
Thus, we need to solve:
\[ (m + 2)(2m + 1) > 0 \]

### Step 5: Find the critical points
The critical points occur when the expression equals zero:
\[ m + 2 = 0 \quad \Rightarrow \quad m = -2 \]
\[ 2m + 1 = 0 \quad \Rightarrow \quad m = -\frac{1}{2} \]

### Step 6: Test intervals around the critical points
We will test the intervals determined by the critical points \( m = -2 \) and \( m = -\frac{1}{2} \):
1. For \( m < -2 \): Choose \( m = -3 \)
- \((m + 2)(2m + 1) = (-3 + 2)(2(-3) + 1) = (-1)(-5) > 0\) (Positive)

2. For \( -2 < m < -\frac{1}{2} \): Choose \( m = -1 \)
- \((m + 2)(2m + 1) = (-1 + 2)(2(-1) + 1) = (1)(-1) < 0\) (Negative)

3. For \( m > -\frac{1}{2} \): Choose \( m = 0 \)
- \((m + 2)(2m + 1) = (0 + 2)(2(0) + 1) = (2)(1) > 0\) (Positive)

### Step 7: Conclusion
The intervals where the product is positive are:
\[ m < -2 \quad \text{or} \quad m > -\frac{1}{2} \]

### Final Answer
The values of \( m \) for which the vectors include an acute angle are:
\[ m < -2 \quad \text{or} \quad m > -\frac{1}{2} \]