The values of x for which the angle between the vectors ` veca =xhati - 3hatj-hatk and vecb = 2x hati + x hatj -hatk` is acute, and the angle, between the vector `vecb` and the axis of ordinates is obtuse, are

Explanation

To solve the problem, we need to find the values of \( x \) for which the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is acute, and the angle between the vector \( \vec{b} \) and the axis of ordinates (y-axis) is obtuse.

### Step 1: Define the vectors
Given:
\[ \vec{a} = x \hat{i} - 3 \hat{j} - \hat{k} \]
\[ \vec{b} = 2x \hat{i} + x \hat{j} - \hat{k} \]

### Step 2: Find the dot product of \( \vec{a} \) and \( \vec{b} \)
The dot product \( \vec{a} \cdot \vec{b} \) is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (x)(2x) + (-3)(x) + (-1)(-1) = 2x^2 - 3x + 1 \]

### Step 3: Condition for acute angle
For the angle between the vectors \( \vec{a} \) and \( \vec{b} \) to be acute, the dot product must be greater than zero:
\[ 2x^2 - 3x + 1 > 0 \]

### Step 4: Solve the quadratic inequality
First, we find the roots of the equation \( 2x^2 - 3x + 1 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} = \frac{3 \pm \sqrt{9 - 8}}{4} = \frac{3 \pm 1}{4} \]
This gives us the roots:
\[ x = 1 \quad \text{and} \quad x = \frac{1}{2} \]

### Step 5: Test intervals for the inequality
We need to test the intervals determined by the roots \( \frac{1}{2} \) and \( 1 \):
- For \( x < \frac{1}{2} \), choose \( x = 0 \):
\[ 2(0)^2 - 3(0) + 1 = 1 > 0 \quad \text{(True)} \]
- For \( \frac{1}{2} < x < 1 \), choose \( x = \frac{3}{4} \):
\[ 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 1 = 2\left(\frac{9}{16}\right) - \frac{9}{4} + 1 = \frac{18}{16} - \frac{36}{16} + \frac{16}{16} = -\frac{2}{16} < 0 \quad \text{(False)} \]
- For \( x > 1 \), choose \( x = 2 \):
\[ 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3 > 0 \quad \text{(True)} \]

Thus, the solution for the first condition is:
\[ x < \frac{1}{2} \quad \text{or} \quad x > 1 \]

### Step 6: Condition for obtuse angle with the y-axis
The angle between \( \vec{b} \) and the y-axis is obtuse if the dot product \( \vec{b} \cdot \hat{j} < 0 \):
\[ \vec{b} \cdot \hat{j} = x < 0 \]

### Step 7: Combine the conditions
From the first condition, we have \( x < \frac{1}{2} \) or \( x > 1 \). From the second condition, we have \( x < 0 \).

The common solution is:
\[ x < 0 \]

### Final Answer
The values of \( x \) for which the angle between the vectors \( \vec{a} \) and \( \vec{b} \) is acute, and the angle between the vector \( \vec{b} \) and the axis of ordinates is obtuse, are:
\[ x < 0 \]