The values of `lambda` for which the angle between the vectors ` a= lambdai - 3j-k` and `b =2lambdai+lambdaj-k` is acute and the angle between b and y-axis lies between `pi//2` and `pi` are

Options

A. `-1`
B. all `lambda gt 0`
C. 1
D. all `lambda lt 0`

Answer

To solve the problem, we need to find the values of \( \lambda \) for which the angle between the vectors \( \mathbf{a} = \lambda \mathbf{i} - 3 \mathbf{j} - \mathbf{k} \) and \( \mathbf{b} = 2\lambda \mathbf{i} + \lambda \mathbf{j} - \mathbf{k} \) is acute, and the angle between \( \mathbf{b} \) and the y-axis lies between \( \frac{\pi}{2} \) and \( \pi \).

### Step 1: Determine the condition for the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \) to be acute.
The angle between two vectors is acute if their dot product is positive. Therefore, we need to calculate \( \mathbf{a} \cdot \mathbf{b} \) and set it greater than zero.

\[ \mathbf{a} \cdot \mathbf{b} = (\lambda \mathbf{i} - 3 \mathbf{j} - \mathbf{k}) \cdot (2\lambda \mathbf{i} + \lambda \mathbf{j} - \mathbf{k}) \]

Calculating the dot product:

\[ \mathbf{a} \cdot \mathbf{b} = \lambda \cdot 2\lambda + (-3) \cdot \lambda + (-1) \cdot (-1) \]

\[ = 2\lambda^2 - 3\lambda + 1 \]

Setting the dot product greater than zero:

\[ 2\lambda^2 - 3\lambda + 1 > 0 \]

### Step 2: Solve the quadratic inequality.
To solve the inequality \( 2\lambda^2 - 3\lambda + 1 > 0 \), we first find the roots of the corresponding equation:

\[ 2\lambda^2 - 3\lambda + 1 = 0 \]

Using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ \lambda = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]

\[ = \frac{3 \pm \sqrt{9 - 8}}{4} \]

\[ = \frac{3 \pm 1}{4} \]

The roots are:

\[ \lambda_1 = 1 \quad \text{and} \quad \lambda_2 = \frac{1}{2} \]

### Step 3: Analyze the intervals.
The roots divide the number line into intervals: \( (-\infty, \frac{1}{2}) \), \( (\frac{1}{2}, 1) \), and \( (1, \infty) \).

We test a point from each interval to determine where the inequality holds:

1. For \( \lambda < \frac{1}{2} \) (e.g., \( \lambda = 0 \)):
\[ 2(0)^2 - 3(0) + 1 = 1 > 0 \]

2. For \( \frac{1}{2} < \lambda < 1 \) (e.g., \( \lambda = 0.75 \)):
\[ 2(0.75)^2 - 3(0.75) + 1 = 1.125 - 2.25 + 1 = -0.125 < 0 \]

3. For \( \lambda > 1 \) (e.g., \( \lambda = 2 \)):
\[ 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3 > 0 \]

The inequality \( 2\lambda^2 - 3\lambda + 1 > 0 \) holds for:

\[ \lambda \in (-\infty, \frac{1}{2}) \cup (1, \infty) \]

### Step 4: Condition for the angle between \( \mathbf{b} \) and the y-axis.
The angle between \( \mathbf{b} \) and the y-axis is between \( \frac{\pi}{2} \) and \( \pi \) if \( \mathbf{b} \cdot \mathbf{j} < 0 \).

Calculating \( \mathbf{b} \cdot \mathbf{j} \):

\[ \mathbf{b} \cdot \mathbf{j} = (2\lambda \mathbf{i} + \lambda \mathbf{j} - \mathbf{k}) \cdot \mathbf{j} = \lambda \]

Setting the condition:

\[ \lambda < 0 \]

### Step 5: Combine the conditions.
We have two conditions:
1. \( \lambda \in (-\infty, \frac{1}{2}) \cup (1, \infty) \)
2. \( \lambda < 0 \)

The intersection of these conditions is:

\[ \lambda \in (-\infty, 0) \]

### Final Answer:
The values of \( \lambda \) for which the conditions are satisfied are:

\[ \lambda < 0 \]