3d Geometry

Step 1: Write in standard form. First line passes through , direction vector . Second line passes through , direction vector . Step 2: Check relation of lines. Since both lines have the same direction vector , they are parallel. Step 3: Formula for shortest distance between parallel lines. Step 4: Find . Step 5: Compute cross product. Step 6: Magnitudes. Step 7: Distance.

Final Answer:

Step 1: Compare with symmetric form. A line in symmetric form is: Here, So,

Step 2: Write vector equation of line. General form: where is position vector of a point, is direction vector.

Step 3: Substitute values. So,

Final Answer:

Step 1: Equation of required plane. The equation of a plane passing through the line of intersection of two planes is: Here, So,

Step 2: General equation of the plane.

Step 3: Condition of passing through . Substitute :

Step 4: Required plane. Substitute : Multiply through by :

Final Answer:

Step 1: Identify points and direction vectors. - Point on line 1: - Direction vector of line 1: - Point on line 2: - Direction vector of line 2: Step 2: Formula for shortest distance. where . Step 3: Compute cross product. So, . Step 4: Compute numerator. So, . Step 5: Compute denominator. Step 6: Final result.

Final Answer:

Step 1: Recall formula. The equation of a plane passing through point with normal vector is:

Step 2: Write given data. Normal vector: Point:

Step 3: Vector equation of plane.

Step 4: Write both forms. - Vector form: - Cartesian form:

Final Answer: Vector equation: Cartesian equation:

Step 1: Write plane equation in Cartesian form.

Step 2: Distance formula.
For a point and plane , Here, , , , , and point .

Step 3: Substitute values.

Final Answer:

To find the shortest distance between two skew lines, we use the formula:

where:

• and are position vectors of any points on the two lines.
• and are the direction vectors of the two lines.

From the given equations:

First, we compute :

Now, compute the magnitude:

Next, compute :

Now, compute the dot product:

Finally, compute the shortest distance: