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 parametric form
The equations can be rewritten in terms of a parameter \(t\).
For the first line \(2x = 3y = -z\):
Let \(2x = k\), then:
- \(x = \frac{k}{2}\)
- \(3y = k \Rightarrow y = \frac{k}{3}\)
- \(-z = k \Rightarrow z = -k\)
Thus, we can express the first line as:
\[
\mathbf{r_1} = \left(\frac{k}{2}, \frac{k}{3}, -k\right)
\]
For the second line \(6x = -y = -4z\):
Let \(6x = m\), then:
- \(x = \frac{m}{6}\)
- \(-y = m \Rightarrow y = -m\)
- \(-4z = m \Rightarrow z = -\frac{m}{4}\)
Thus, we can express the second line as:
\[
\mathbf{r_2} = \left(\frac{m}{6}, -m, -\frac{m}{4}\right)
\]
### Step 2: Identify direction ratios
From the parametric equations, we can identify the direction ratios of the lines.
For the first line, the direction ratios are:
\[
\mathbf{a_1} = \left(\frac{1}{2}, \frac{1}{3}, -1\right)
\]
Multiplying by 6 to eliminate fractions, we get:
\[
\mathbf{a_1} = (3, 2, -6)
\]
For the second line, the direction ratios are:
\[
\mathbf{a_2} = \left(\frac{1}{6}, -1, -\frac{1}{4}\right)
\]
Multiplying by 12 to eliminate fractions, we get:
\[
\mathbf{a_2} = (2, -12, -3)
\]
### Step 3: Use the formula for the angle between two lines
The angle \(\theta\) between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is given by:
\[
\cos \theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}
\]
### Step 4: Substitute the values
Substituting the values:
- \(a_1 = 3\), \(b_1 = 2\), \(c_1 = -6\)
- \(a_2 = 2\), \(b_2 = -12\), \(c_2 = -3\)
Calculating the numerator:
\[
|3 \cdot 2 + 2 \cdot (-12) + (-6) \cdot (-3)| = |6 - 24 + 18| = |0| = 0
\]
Calculating the denominator:
\[
\sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7
\]
\[
\sqrt{2^2 + (-12)^2 + (-3)^2} = \sqrt{4 + 144 + 9} = \sqrt{157}
\]
Thus, we have:
\[
\cos \theta = \frac{0}{7 \cdot \sqrt{157}} = 0
\]
### Step 5: Find the angle
Since \(\cos \theta = 0\), it follows that:
\[
\theta = \cos^{-1}(0) = \frac{\pi}{2} \text{ radians} = 90^\circ
\]
### Conclusion
The angle between the lines is \(90^\circ\).
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