The angle between the lines => `2x=3y=-z and -6x=y=4z` is:

Options

A. `30^(@)`
B. `45^(@)`
C. `90^(@)`
D. `0^(@)`

Answer

To find the angle between the lines given by the equations \(2x = 3y = -z\) and \(-6x = y = 4z\), we will follow these steps:

### Step 1: Convert the equations into standard form
The standard form of a line in three-dimensional space is given by:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
where \((x_1, y_1, z_1)\) is a point on the line and \((a, b, c)\) are the direction ratios of the line.

#### For Line 1: \(2x = 3y = -z\)
1. Set \(2x = 3y = -z = k\).
- From \(2x = k\), we have \(x = \frac{k}{2}\).
- From \(3y = k\), we have \(y = \frac{k}{3}\).
- From \(-z = k\), we have \(z = -k\).

2. Therefore, the parametric equations are:
\[ x = \frac{k}{2}, \quad y = \frac{k}{3}, \quad z = -k \]

3. The direction ratios for Line 1 are \((2, 3, -1)\).

#### For Line 2: \(-6x = y = 4z\)
1. Set \(-6x = y = 4z = m\).
- From \(-6x = m\), we have \(x = -\frac{m}{6}\).
- From \(y = m\), we have \(y = m\).
- From \(4z = m\), we have \(z = \frac{m}{4}\).

2. Therefore, the parametric equations are:
\[ x = -\frac{m}{6}, \quad y = m, \quad z = \frac{m}{4} \]

3. The direction ratios for Line 2 are \((-6, 1, 4)\).

### Step 2: Find the angle between the two lines
The angle \(\theta\) between two lines can be found using the formula:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]
where \(\mathbf{A}\) and \(\mathbf{B}\) are the direction ratios of the two lines.

1. Let \(\mathbf{A} = (2, 3, -1)\) and \(\mathbf{B} = (-6, 1, 4)\).

2. Calculate the dot product \(\mathbf{A} \cdot \mathbf{B}\):
\[ \mathbf{A} \cdot \mathbf{B} = 2 \cdot (-6) + 3 \cdot 1 + (-1) \cdot 4 = -12 + 3 - 4 = -13 \]

3. Calculate the magnitudes:
\[ |\mathbf{A}| = \sqrt{2^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14} \]
\[ |\mathbf{B}| = \sqrt{(-6)^2 + 1^2 + 4^2} = \sqrt{36 + 1 + 16} = \sqrt{53} \]

4. Substitute into the cosine formula:
\[ \cos \theta = \frac{-13}{\sqrt{14} \cdot \sqrt{53}} \]

5. To find the angle \(\theta\), we can use the fact that if \(\cos \theta = 0\), then \(\theta = 90^\circ\). In this case, we will find that the dot product is negative, indicating that the angle is obtuse.

### Conclusion
Since the cosine of the angle is negative, the angle between the two lines is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

### Final Answer
The angle between the lines is \(90^\circ\).