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

Options

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

Answer

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

### Step 1: Write the equations in symmetric form
The first line \(2x = 3y = -z\) can be rewritten in symmetric form. We can express it as:
\[ \frac{x}{3} = \frac{y}{2} = \frac{z}{-6} \]
This gives us the direction ratios for the first line as \( (3, 2, -6) \).

### Step 2: Write the second line in symmetric form
The second line \(6x = -y = -4z\) can also be rewritten in symmetric form:
\[ \frac{x}{2} = \frac{y}{-12} = \frac{z}{-3} \]
This gives us the direction ratios for the second line as \( (2, -12, -3) \).

### Step 3: Identify the direction ratios
From the above steps, we have:
- Direction ratios of the first line: \( \mathbf{a_1} = (3, 2, -6) \)
- Direction ratios of the second line: \( \mathbf{a_2} = (2, -12, -3) \)

### Step 4: Use the formula for the angle between two lines
The angle \( \theta \) between two lines can be calculated using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]
Substituting the values:
- \( a_1 = 3, b_1 = 2, c_1 = -6 \)
- \( a_2 = 2, b_2 = -12, c_2 = -3 \)

### Step 5: Calculate the numerator
Calculating the numerator:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = (3)(2) + (2)(-12) + (-6)(-3) \]
\[ = 6 - 24 + 18 = 0 \]

### Step 6: Calculate the denominator
Calculating the denominator:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{2^2 + (-12)^2 + (-3)^2} = \sqrt{4 + 144 + 9} = \sqrt{157} \]

### Step 7: Substitute into the formula
Now substituting back into the formula:
\[ \cos \theta = \frac{0}{7 \cdot \sqrt{157}} = 0 \]

### Step 8: Determine the angle
Since \( \cos \theta = 0 \), it implies:
\[ \theta = \frac{\pi}{2} \text{ or } 90^\circ \]

### Conclusion
The angle between the lines \(2x = 3y = -z\) and \(6x = -y = -4z\) is \(90^\circ\).

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