Three Dimensional Geometry

Three points are collinear if the vectors formed by them are parallel.
Let us consider:

Since the vectors and are in the same direction (i.e., collinear), one must be a scalar multiple of the other: Comparing components:

Now solve the equation:

Final Answer:

Intersection of Two Lines in 3D

Step 1: Represent the lines in vector form

Line 1 (from symmetric form):

Line 2 (from symmetric form):

Step 2: Assume the lines intersect

If the lines intersect, then:

Equating components:

• (i)
• (ii)
• (iii)

Step 3: Solve the system

From (iii): Substitute into (i):

Now, from (iii):

Check in (ii):

✅ Final Answer:

The lines intersect at the point:

Given Vector Equations of Lines

Line 1: Line 2:

Step 1: Direction Vectors

Since the direction vectors are scalar multiples ⇒ lines are parallel.

Step 2: Check for Coincidence

Take position vectors of a point on each line: Compute: Check if this is a scalar multiple of : It is not, so lines are parallel but not coincident.

Step 3: Find Shortest Distance Between Lines

Use formula: Let: Compute cross product:

✅ Final Conclusion:

• Lines are parallel
• They are not coincident
• Shortest distance between them is approximately:

Find the Vector Equation of a Line

Given:

We are to find the vector equation of a line passing through the point and perpendicular to both the lines:

Line 1: Line 2:

Step 1: Extract Direction Vectors

Direction vector of Line 1: Direction vector of Line 2:

Step 2: Find Cross Product

The required line is perpendicular to both lines, so we compute the cross product:

Step 3: Use Point (1, 2, -3)

Line passes through point

Step 4: Vector Equation of the Required Line

Step 5: Parametric Form

✅ Final Answer:

The vector equation is: