The angle between the lines passing through the points (8, 2, 0), (4, 6, -7) and (-3, 1, 2), (-9, -2, 4) is

Options

A. `cos^(-1)((2)/(63))`
B. `cos^(-1)((20)/(63))`
C. `(pi)/(2)`
D. `(pi)/(5)`

Answer

To find the angle between the lines passing through the points (8, 2, 0), (4, 6, -7) and (-3, 1, 2), (-9, -2, 4), we will follow these steps:

### Step 1: Find the direction ratios of the lines.

For the first line passing through points \( A(8, 2, 0) \) and \( B(4, 6, -7) \):

- Direction ratios \( \text{DR}_1 = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)
- Here, \( x_1 = 8, y_1 = 2, z_1 = 0 \) and \( x_2 = 4, y_2 = 6, z_2 = -7 \)

Calculating the direction ratios:
\[ \text{DR}_1 = (4 - 8, 6 - 2, -7 - 0) = (-4, 4, -7) \]

For the second line passing through points \( C(-3, 1, 2) \) and \( D(-9, -2, 4) \):

- Direction ratios \( \text{DR}_2 = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)
- Here, \( x_1 = -3, y_1 = 1, z_1 = 2 \) and \( x_2 = -9, y_2 = -2, z_2 = 4 \)

Calculating the direction ratios:
\[ \text{DR}_2 = (-9 + 3, -2 - 1, 4 - 2) = (-6, -3, 2) \]

### Step 2: Use the direction ratios to find the cosine of the angle between the lines.

Let \( \text{DR}_1 = (a_1, b_1, c_1) = (-4, 4, -7) \) and \( \text{DR}_2 = (a_2, b_2, c_2) = (-6, -3, 2) \).

The formula for the cosine of the angle \( \theta \) between two lines is given by:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

Calculating the numerator:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = (-4)(-6) + (4)(-3) + (-7)(2) \]
\[ = 24 - 12 - 14 = -2 \]

Calculating the denominator:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{(-4)^2 + 4^2 + (-7)^2} = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{(-6)^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7 \]

Putting it all together:
\[ \cos \theta = \frac{-2}{9 \times 7} = \frac{-2}{63} \]

### Step 3: Find the angle \( \theta \).

To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{-2}{63}\right) \]

### Final Answer:
The angle between the lines is \( \theta = \cos^{-1}\left(\frac{-2}{63}\right) \).

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