If A,B,C,D are (2,3,-1),(3,5,-3),(1,2,3),(3,5,7) respectively, then the angel between AB and CD, is

Options

A. `(pi)/2`
B. `(pi)/3`
C. `(pi)/4`
D. `(pi)/6`

Explanation

To find the angle between the lines defined by points A, B, C, and D, we will follow these steps:

### Step 1: Determine the position vectors of points A and B
Given the coordinates of points A and B:
- A = (2, 3, -1)
- B = (3, 5, -3)

The vector **AB** can be calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} = (3 - 2, 5 - 3, -3 - (-1)) = (1, 2, -2) \]

### Step 2: Determine the position vectors of points C and D
Given the coordinates of points C and D:
- C = (1, 2, 3)
- D = (3, 5, 7)

The vector **CD** can be calculated as:
\[ \vec{CD} = \vec{D} - \vec{C} = (3 - 1, 5 - 2, 7 - 3) = (2, 3, 4) \]

### Step 3: Calculate the dot product of vectors AB and CD
The dot product \(\vec{AB} \cdot \vec{CD}\) is calculated as follows:
\[ \vec{AB} \cdot \vec{CD} = (1)(2) + (2)(3) + (-2)(4) = 2 + 6 - 8 = 0 \]

### Step 4: Calculate the magnitudes of vectors AB and CD
The magnitude of vector **AB** is:
\[ |\vec{AB}| = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

The magnitude of vector **CD** is:
\[ |\vec{CD}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

### Step 5: Use the dot product to find cos(θ)
Using the formula for the cosine of the angle between two vectors:
\[ \cos \theta = \frac{\vec{AB} \cdot \vec{CD}}{|\vec{AB}| \cdot |\vec{CD}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{0}{3 \cdot \sqrt{29}} = 0 \]

### Step 6: Determine the angle θ
The cosine of the angle is zero when:
\[ \theta = 90^\circ \quad \text{or} \quad \theta = \frac{\pi}{2} \text{ radians} \]

### Conclusion
The angle between the lines AB and CD is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

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