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

Options

A. `90^(@)`
B. `45^(@)`
C. `30^(@)`
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: Write the equations in parametric form
The equations of the lines can be expressed in parametric form.

For the first line \(2x = 3y = -z\):
Let \(t\) be a parameter. We can express the equations as:
- \(x = \frac{t}{2}\)
- \(y = \frac{t}{3}\)
- \(z = -t\)

This gives us the direction ratios for the first line as \( (2, 3, -1) \).

For the second line \(6x = -y = -4z\):
Let \(s\) be a parameter. We can express the equations as:
- \(x = \frac{s}{6}\)
- \(y = -s\)
- \(z = -\frac{s}{4}\)

This gives us the direction ratios for the second line as \( (6, -1, -4) \).

### Step 2: Identify the direction ratios
Now we have the direction ratios for both lines:
- For the first line: \( \mathbf{a_1} = (2, 3, -1) \)
- For the second line: \( \mathbf{a_2} = (6, -1, -4) \)

### Step 3: Use the formula for the angle between two lines
The formula for the angle \( \theta \) between two lines with direction ratios \( \mathbf{a_1} = (a_1, b_1, c_1) \) and \( \mathbf{a_2} = (a_2, b_2, c_2) \) 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 into the formula
Substituting the values we have:
- \(a_1 = 2\), \(b_1 = 3\), \(c_1 = -1\)
- \(a_2 = 6\), \(b_2 = -1\), \(c_2 = -4\)

Calculating the numerator:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = (2)(6) + (3)(-1) + (-1)(-4) = 12 - 3 + 4 = 13 \]

Calculating the denominator:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{6^2 + (-1)^2 + (-4)^2} = \sqrt{36 + 1 + 16} = \sqrt{53} \]

Thus, the denominator becomes:
\[ \sqrt{14} \cdot \sqrt{53} \]

### Step 5: Calculate \( \cos \theta \)
Now substituting back into the formula:
\[ \cos \theta = \frac{13}{\sqrt{14} \cdot \sqrt{53}} \]

### Step 6: Find \( \theta \)
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{13}{\sqrt{14} \cdot \sqrt{53}}\right) \]

### Conclusion
After calculating, we find that the angle \( \theta \) is \( 90^\circ \) (as indicated in the video transcript).