各直線はN次元空間のベクトルと点によって定義される：

```python
import numpy as np

def line_intersection(point1, vector1, point2, vector2):
    # Convert inputs to numpy arrays
    point1 = np.array(point1)
    vector1 = np.array(vector1)
    point2 = np.array(point2)
    vector2 = np.array(vector2)
    
    # Calculate the intersection point
    A = np.vstack((vector1, -vector2)).T
    b = point2 - point1
    x = np.linalg.solve(A, b)
    intersection = point1 + vector1 * x[0]
    
    return intersection
```

line_intersection` 関数は4つの引数を取る：point1`, `vector1`, `point2`, `vector2` である。各点は N 次元空間の点を表す長さ N のリストまたはタプルである。各ベクトルは直線の方向を表す長さ N のリストまたはタプルである。

この関数は，計算を簡単にするために，まず入力を numpy の配列に変換します．そして、交点を求める一次方程式系を構築します。連立方程式は

```
x1 * v1_1 + x2 * v1_2 + ... + xN * v1_N = p2_1 - p1_1
x1 * v2_1 + x2 * v2_2 + ... + xN * v2_N = p2_2 - p1_2
```

ここで、`p1`と`v1`は1本目の線の点とベクトルを表し、`p2`と`v2`は2本目の線の点とベクトルを表す。x1`、`x2`、...、`xN`は交点の成分である。

次に、この関数は、numpy の `linalg.solve` 関数を使用してこの連立方程式を解き、 `x1`, `x2`, ..., `xN` の値を求めます。最後に、この関数は交点を `p1 + x1 * v1` として計算します。

この関数の使用例を以下に示す：

```python
point1 = [1, 2, 3]
vector1 = [2, 3, 1]
point2 = [3, 1, 2]
vector2 = [1, 2, 3]

intersection = line_intersection(point1, vector1, point2, vector2)
print(intersection)
```

これはN次元空間における2本の直線の交点を出力する。
Here is a Python program that finds the intersection between two lines, where each line is defined by a vector and a point in N-dimensional space:

```python
import numpy as np

def line_intersection(point1, vector1, point2, vector2):
    # Convert inputs to numpy arrays
    point1 = np.array(point1)
    vector1 = np.array(vector1)
    point2 = np.array(point2)
    vector2 = np.array(vector2)
    
    # Calculate the intersection point
    A = np.vstack((vector1, -vector2)).T
    b = point2 - point1
    x = np.linalg.solve(A, b)
    intersection = point1 + vector1 * x[0]
    
    return intersection
```

The `line_intersection` function takes four arguments: `point1`, `vector1`, `point2`, and `vector2`. Each point is a list or tuple of length N representing a point in N-dimensional space. Each vector is a list or tuple of length N representing the direction of the line.

The function first converts the inputs to numpy arrays to make it easier to perform calculations. It then constructs a system of linear equations to solve for the intersection point. The system of equations is:

```
x1 * v1_1 + x2 * v1_2 + ... + xN * v1_N = p2_1 - p1_1
x1 * v2_1 + x2 * v2_2 + ... + xN * v2_N = p2_2 - p1_2
```

where `p1` and `v1` represent the point and vector of the first line, and `p2` and `v2` represent the point and vector of the second line. `x1`, `x2`, ..., `xN` are the components of the intersection point.

The function then solves this system of equations using numpy's `linalg.solve` function to find the values of `x1`,` x2`, ..., `xN`. Finally, the function calculates the intersection point as `p1 + x1 * v1`.

Here's an example usage of the function:

```python
point1 = [1, 2, 3]
vector1 = [2, 3, 1]
point2 = [3, 1, 2]
vector2 = [1, 2, 3]

intersection = line_intersection(point1, vector1, point2, vector2)
print(intersection)
```

This should output the intersection point between the two lines in N-dimensional space.