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

### Step 1: Rewrite the equations in symmetric form
The first line can be expressed as:
\[ \frac{x}{\frac{1}{2}} = \frac{y}{\frac{1}{3}} = \frac{z}{-1} \]
This gives us the direction ratios of the first line as \( (1/2, 1/3, -1) \).

The second line can be expressed as:
\[ \frac{x}{\frac{1}{6}} = \frac{y}{-1} = \frac{z}{-\frac{1}{4}} \]
This gives us the direction ratios of the second line as \( (1/6, -1, -1/4) \).

### Step 2: Scale the direction ratios
To simplify calculations, we can scale the direction ratios of both lines.

For the first line, multiplying by 6 gives:
\[ (3, 2, -6) \]

For the second line, multiplying by 12 gives:
\[ (2, -12, -3) \]

### 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 \( (a_2, b_2, c_2) \) can be found using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 4: Substitute the values
Substituting the values we found:
- For the first line: \( a_1 = 3, b_1 = 2, c_1 = -6 \)
- For the second line: \( a_2 = 2, b_2 = -12, c_2 = -3 \)

Calculating the numerator:
\[ 3 \cdot 2 + 2 \cdot (-12) + (-6) \cdot (-3) = 6 - 24 + 18 = 0 \]

Calculating the denominator:
\[ \sqrt{3^2 + 2^2 + (-6)^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]
\[ \sqrt{2^2 + (-12)^2 + (-3)^2} = \sqrt{4 + 144 + 9} = \sqrt{157} \]

### Step 5: Calculate \( \cos \theta \)
Now substituting back into the formula:
\[ \cos \theta = \frac{0}{7 \cdot \sqrt{157}} = 0 \]

### Step 6: Determine the angle
Since \( \cos \theta = 0 \), this implies:
\[ \theta = 90^\circ \]

### Conclusion
The measure of the angle between the two lines is \( 90^\circ \).

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