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 AB and CD, we will follow these steps:

### Step 1: Find the direction vectors of AB and CD.

1. **Calculate the direction vector of AB**:
- Points A and B are given as A(2, 3, -1) and B(3, 5, -3).
- The direction vector AB = B - A = (3 - 2, 5 - 3, -3 - (-1)) = (1, 2, -2).

2. **Calculate the direction vector of CD**:
- Points C and D are given as C(1, 2, 3) and D(3, 5, 7).
- The direction vector CD = D - C = (3 - 1, 5 - 2, 7 - 3) = (2, 3, 4).

### Step 2: Use the formula for the angle between two vectors.

The angle θ between two vectors **u** and **v** can be found using the formula:
\[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \]
where:
- **u** = AB = (1, 2, -2)
- **v** = CD = (2, 3, 4)

### Step 3: Calculate the dot product of AB and CD.

\[ \mathbf{u} \cdot \mathbf{v} = (1)(2) + (2)(3) + (-2)(4) = 2 + 6 - 8 = 0 \]

### Step 4: Calculate the magnitudes of the vectors AB and CD.

1. **Magnitude of AB**:
\[ |\mathbf{u}| = \sqrt{(1)^2 + (2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

2. **Magnitude of CD**:
\[ |\mathbf{v}| = \sqrt{(2)^2 + (3)^2 + (4)^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

### Step 5: Substitute values into the cosine formula.

\[ \cos(\theta) = \frac{0}{3 \cdot \sqrt{29}} = 0 \]

### Step 6: Find the angle θ.

Since \(\cos(\theta) = 0\), this implies:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between the lines AB and CD is \(90^\circ\).

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