3d Geometry

Step 1: Find Three Vectors

Step 2: Compute the Volume

Final Answer: .

Step 1: Identify the Direction and Normal Vectors
- The direction vector of the line is .
- The normal vector of the plane is .
Step 2: Apply the Formula for the Angle Between Line and Plane
The angle between the line and the plane can be calculated using the angle between the direction vector of the line and the normal vector of the plane:

We are given two skew lines, whose equations are:
Step 1: Identify points and direction vectors.
The lines pass through the points: and are parallel to the direction vectors:
Step 2: Find the vector between the points.
The vector between the two points and is:
Step 3: Calculate the cross product of the direction vectors.
Now, calculate the cross product of the direction vectors and :

Step 4: Apply the shortest distance formula.
The formula for the shortest distance between two skew lines is given by: Substituting the values:
Final Answer: Thus, the shortest distance between the two lines is: