3d Geometry

Step 1: Use substitution:
Step 2: Substitute in the integral:
Step 3: Integrate using the standard result:
Step 4: Therefore,

Step 1: Let
Step 2: Replace by in the integral:
Step 3: Add the two expressions for :
Step 4: Take outside the integral:
Step 5: Substitute , so . When , ; when , .
Step 6: Hence,

Step 1: Use 3D section formula. Let ratio .
Step 2: Point coordinates of division:
Step 3: Substitute in plane equation and solve linear relation → gives 3:7. Hence → (C).

Step 1: Use point–plane distance:
Step 2: For first point: For second:
Step 3: Hence → (C).

Step 1: Recall the definition of a unit vector. A vector is a unit vector if its magnitude is equal to . Step 2: Find the magnitude of the given vector. Given vector: Magnitude: Step 3: Apply the unit vector condition. Step 4: Write both possible values of . Step 5: Final conclusion. The correct answer is:

Step 1: Recall the concept. A line perpendicular to two given lines has direction ratios equal to the cross product of their direction ratios. Step 2: Write the given direction ratios. Step 3: Find the cross product . Step 4: Convert direction ratios into direction cosines. Magnitude: Direction cosines: The opposite direction is also valid: Step 5: Match with the given options. Option (C) matches one valid set of direction cosines. Final Answer: