3d Geometry

The distance of a point from a plane is given by the formula:

For the point and the plane , the plane equation is:
.
Substitute the point into the distance formula:


Simplifying the numerator:

Step 1: Find the midpoint of segment
The midpoint of the segment joining and is given by: Substituting values: Step 2: Find equation of plane passing through
Since the required plane is parallel to the given plane , its equation follows the same normal vector: Since this plane passes through , substituting these values: Rearranging: Step 3: Compute
Given that the equation satisfies the conditions and substituting appropriate values from the plane equation, we get: Step 4: Conclusion
Thus, the final answer is:

Step 1: Find Direction Ratios of Line
The direction cosines of line are given by: Using the identity: Thus, the direction ratios of are proportional to: Step 2: Find the Angle Between the Two Lines
The direction ratios of the second line are . The angle between the lines is given by: Substituting values: Step 3: Conclusion
Thus, the correct answer is:

Step 1: Condition for perpendicularity
If a line is perpendicular to two given lines with direction ratios and , then its direction cosines satisfy: For the given lines: 1st line: , so 2nd line: , so Step 2: Solving for
Adding the two equations: Substituting in the first equation: Using the direction cosine condition: Step 3: Computing
Step 4: Conclusion
Thus, the final answer is:

To find the length of the internal bisector of angle in with vertices , , and , follow these steps: 1. Calculate the lengths of sides and :
- Length of :
- Length of : 2. Use the Angle Bisector Theorem: The internal bisector of angle divides the opposite side in the ratio of the adjacent sides and . Let the bisector intersect at point . Then: Therefore, . 3. Find the coordinates of point : The coordinates of can be found using the section formula. Given and , and : 4. Calculate the length of the angle bisector : The length of is the distance between and : However, let's verify the calculation: But is not a perfect square, so let's re
-examine the problem. 5. Re
-evaluate the calculation: The correct approach is to use the formula for the length of the angle bisector in a triangle: However, a simpler formula for the length of the internal bisector is: First, calculate : Then: Final Answer: \boxed{\frac{2}{3} \sqrt{34}}