If the angle between the vector `vec a = alpha t hat i + 6 hat j – 3 hat k` and `vec b = t hat i – 2 hat j – 2 alpha t hat k` is obtuse for `t in R` then `alpha` is

Options

A. `(0, 4/3)`
B. `(-4/3, 0]`
C. `(-2, 0)`
D. `(- 3/5, 4/3)`

Answer

To solve the problem, we need to determine the values of \( \alpha \) for which the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is obtuse. The vectors are given as:

\[ \vec{a} = \alpha \hat{i} + 6 \hat{j} - 3 \hat{k} \]
\[ \vec{b} = \hat{i} - 2 \hat{j} - 2\alpha \hat{k} \]

### Step 1: Understand the condition for obtuse angle
The angle \( \theta \) between two vectors is obtuse if the dot product \( \vec{a} \cdot \vec{b} < 0 \).

### Step 2: Calculate the dot product
The dot product \( \vec{a} \cdot \vec{b} \) is calculated as follows:

\[ \vec{a} \cdot \vec{b} = (\alpha \hat{i} + 6 \hat{j} - 3 \hat{k}) \cdot (\hat{i} - 2 \hat{j} - 2\alpha \hat{k}) \]

Calculating this gives:

\[ \vec{a} \cdot \vec{b} = \alpha \cdot 1 + 6 \cdot (-2) + (-3) \cdot (-2\alpha) \]

Simplifying this:

\[ = \alpha - 12 + 6\alpha = 7\alpha - 12 \]

### Step 3: Set up the inequality
For the angle to be obtuse, we need:

\[ 7\alpha - 12 < 0 \]

### Step 4: Solve the inequality
Rearranging the inequality:

\[ 7\alpha < 12 \]
\[ \alpha < \frac{12}{7} \]

### Step 5: Consider the discriminant condition
The dot product leads us to a quadratic equation in terms of \( t \):

\[ \alpha t^2 + 6\alpha t - 12 < 0 \]

For this quadratic to be negative for all \( t \), its discriminant must be less than zero. The discriminant \( D \) is given by:

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

Setting the discriminant less than zero:

\[ 36\alpha^2 + 48\alpha < 0 \]

### Step 6: Factor out common terms
Factoring out \( 12\alpha \):

\[ 12\alpha(3\alpha + 4) < 0 \]

### Step 7: Analyze the factors
The inequality \( 12\alpha(3\alpha + 4) < 0 \) holds when:

1. \( \alpha < 0 \) and \( 3\alpha + 4 > 0 \) (which gives \( \alpha > -\frac{4}{3} \))
2. \( \alpha > 0 \) and \( 3\alpha + 4 < 0 \) (which is not possible since \( \alpha > 0 \) contradicts \( 3\alpha + 4 < 0 \))

Thus, the valid range for \( \alpha \) is:

\[ -\frac{4}{3} < \alpha < 0 \]

### Step 8: Combine inequalities
From the previous steps, we have:

1. \( \alpha < \frac{12}{7} \)
2. \( -\frac{4}{3} < \alpha < 0 \)

The final range for \( \alpha \) is:

\[ -\frac{4}{3} < \alpha < 0 \]

### Final Answer
Thus, the values of \( \alpha \) for which the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is obtuse are:

\[ \alpha \in \left(-\frac{4}{3}, 0\right) \]