how to find intersection of two lines in 3d

To find the intersection of two lines in 3D, the following steps are used:

### 1. **Parametric Equations of the Lines**:
   The parametric form of a line in 3D is typically given as:

   \[
   \text{Line 1: } \vec{r_1}(t) = \vec{r_{01}} + t \vec{d_1}
   \]
   \[
   \text{Line 2: } \vec{r_2}(s) = \vec{r_{02}} + s \vec{d_2}
   \]
   
   Where:
   - \( \vec{r_{01}} \) and \( \vec{r_{02}} \) are the position vectors of points on Line 1 and Line 2, respectively.
   - \( \vec{d_1} \) and \( \vec{d_2} \) are the direction vectors of Line 1 and Line 2.
   - \( t \) and \( s \) are the parameters that vary along the lines.

### 2. **Set the Parametric Equations Equal**:
   If the two lines intersect, then their positions are equal at the intersection point. So, equate the parametric equations of the two lines:

   \[
   \vec{r_{01}} + t \vec{d_1} = \vec{r_{02}} + s \vec{d_2}
   \]

   This gives three scalar equations (one for each coordinate: \(x\), \(y\), and \(z\)).

### 3. **Solve the System of Equations**:
   You now have a system of three equations with two unknowns (\(t\) and \(s\)). Solve these simultaneous equations for \(t\) and \(s\).

   If a consistent solution exists, it means the lines intersect at a point. Plug the found value of \(t\) (or \(s\)) back into the parametric equation of one of the lines to find the coordinates of the intersection point.

### 4. **Check Consistency**:
   - **Consistent solution**: If there is a unique solution for \(t\) and \(s\), the lines intersect at a point.
   - **No solution**: If the system of equations is inconsistent, the lines do not intersect.
   - **Infinite solutions**: If the equations reduce to a single parameter, the lines are coincident (they lie on top of each other).

### Example:
Let's say Line 1 and Line 2 are represented as:

- Line 1: \( \vec{r_1}(t) = (x_1, y_1, z_1) + t (a_1, b_1, c_1) \)
- Line 2: \( \vec{r_2}(s) = (x_2, y_2, z_2) + s (a_2, b_2, c_2) \)

Set up the equations for \(x\), \(y\), and \(z\):

\[
x_1 + t a_1 = x_2 + s a_2
\]
\[
y_1 + t b_1 = y_2 + s b_2
\]
\[
z_1 + t c_1 = z_2 + s c_2
\]

Solve this system for \(t\) and \(s\), and use these to find the intersection point if one exists.

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### 5 Relative Questions:
1. What happens when two lines in 3D space are parallel but distinct?
2. How can we check if two lines are skew (non-intersecting and non-parallel)?
3. Can the intersection point be computed using vector products like the cross product?
4. What changes in the method if one of the lines is given in vector form instead of parametric form?
5. How do we interpret the case where the direction vectors of both lines are linearly dependent?

### Tip:
When solving systems of equations for 3D line intersections, checking the consistency of your equations early can help avoid redundant calculations.

Learn how to find the intersection of two lines in 3D space by solving parametric equations. This method includes setting up parametric forms of lines, solving the system of equations, and checking for consistency. Suitable for Grades 11-12 and undergraduate students.

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