Let `a=2lambda^(2)i+4lambdaj+k" and "b=7i-2j+lambdak`. The number of values of `lambda` for which angle between a and b is `theta`, where `pi//2 lt theta lt pi` and angle between b and k is `phi " where "0 lt varphi lt pi//6`, is _______

Answer

To solve the problem, we need to analyze the vectors \( \mathbf{a} \) and \( \mathbf{b} \) given by:

\[ \mathbf{a} = 2\lambda^2 \mathbf{i} + 4\lambda \mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 7 \mathbf{i} - 2 \mathbf{j} + \lambda \mathbf{k} \]

We need to find the number of values of \( \lambda \) such that the angle \( \theta \) between \( \mathbf{a} \) and \( \mathbf{b} \) is in the range \( \frac{\pi}{2} < \theta < \pi \) and the angle \( \phi \) between \( \mathbf{b} \) and \( \mathbf{k} \) is in the range \( 0 < \phi < \frac{\pi}{6} \).

### Step 1: Find the cosine of the angle \( \theta \)

The cosine of the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by:

\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]

#### Step 1.1: Calculate \( \mathbf{a} \cdot \mathbf{b} \)

\[ \mathbf{a} \cdot \mathbf{b} = (2\lambda^2)(7) + (4\lambda)(-2) + (1)(\lambda) = 14\lambda^2 - 8\lambda + \lambda = 14\lambda^2 - 7\lambda \]

#### Step 1.2: Calculate \( |\mathbf{a}| \) and \( |\mathbf{b}| \)

\[ |\mathbf{a}| = \sqrt{(2\lambda^2)^2 + (4\lambda)^2 + (1)^2} = \sqrt{4\lambda^4 + 16\lambda^2 + 1} \]

\[ |\mathbf{b}| = \sqrt{(7)^2 + (-2)^2 + (\lambda)^2} = \sqrt{49 + 4 + \lambda^2} = \sqrt{53 + \lambda^2} \]

#### Step 1.3: Substitute into the cosine formula

\[ \cos \theta = \frac{14\lambda^2 - 7\lambda}{\sqrt{4\lambda^4 + 16\lambda^2 + 1} \cdot \sqrt{53 + \lambda^2}} \]

### Step 2: Determine the range for \( \cos \theta \)

Since \( \frac{\pi}{2} < \theta < \pi \), we have:

\[ -1 < \cos \theta < 0 \]

This implies:

\[ -1 < \frac{14\lambda^2 - 7\lambda}{\sqrt{(4\lambda^4 + 16\lambda^2 + 1)(53 + \lambda^2)}} < 0 \]

### Step 3: Solve the inequalities

#### Step 3.1: Solve \( 14\lambda^2 - 7\lambda < 0 \)

Factoring gives:

\[ 7\lambda(2\lambda - 1) < 0 \]

This inequality holds when:

1. \( \lambda < 0 \) (negative)
2. \( 0 < \lambda < \frac{1}{2} \)

#### Step 3.2: Solve \( 14\lambda^2 - 7\lambda > -\sqrt{(4\lambda^4 + 16\lambda^2 + 1)(53 + \lambda^2)} \)

This is more complex and will require numerical or graphical methods to analyze further.

### Step 4: Analyze the angle \( \phi \)

The angle \( \phi \) between \( \mathbf{b} \) and \( \mathbf{k} \) is given by:

\[ \cos \phi = \frac{\mathbf{b} \cdot \mathbf{k}}{|\mathbf{b}|} \]

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

\[ \mathbf{b} \cdot \mathbf{k} = \lambda \]

Thus,

\[ \cos \phi = \frac{\lambda}{\sqrt{53 + \lambda^2}} \]

For \( 0 < \phi < \frac{\pi}{6} \), we have:

\[ \frac{\sqrt{3}}{2} < \frac{\lambda}{\sqrt{53 + \lambda^2}} < 1 \]

### Step 5: Combine conditions

We need to find the values of \( \lambda \) that satisfy both conditions from steps 3 and 4.

### Conclusion

After analyzing both conditions, we find that they contradict each other, leading us to conclude that there are **zero values of \( \lambda \)** that satisfy both conditions.

### Final Answer

The number of values of \( \lambda \) is **0**.

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