3d Geometry

Step 1: Shortest distance between skew lines. The shortest distance between two skew lines is given by the formula: Where and are points on the lines and , are the direction vectors of the lines.
Step 2: Conclusion. The shortest distance between the lines is .

Step 1: Use the method of linear combination. To find the equation of the plane passing through the line of intersection of two planes, we use a linear combination of the given planes' equations, ensuring that the point lies on the plane.
Step 2: Conclusion. Thus, the equation of the required plane is .

Step 1: Volume of tetrahedron. The volume of a tetrahedron with vertices , , , and can be found using the determinant formula.
Step 2: Conclusion. Thus, the volume of the tetrahedron is .

Step 1: Using the formula for the cosine of the angle between two vectors.
The cosine of the angle between two vectors and is given by: After calculating the dot product and magnitudes, the result is .

Step 2: Conclusion.
Thus, the correct answer is option (A).

Final Answer:

Step 1: Use the direction cosines.
The angles with the positive axes give the direction cosines of the line, allowing us to determine the angle with the z-axis.
Step 2: Conclusion.
Thus, the angle with the z-axis is 60°.
Final Answer:

We are given that the plane meets the coordinate axes at points , , and . Let the coordinates of these points be: - on the -axis,
- on the -axis,
- on the -axis.
The equation of the plane passing through the points , , and can be written as: Now, we are given that the centroid of lies at the point . The centroid of a triangle with vertices , , and is given by the average of the coordinates of the vertices:
Substituting the coordinates of , , and , the centroid is: We are told that the centroid is at . Thus, we have the system of equations: Solving these equations gives: Step 2: Equation of the plane. Substitute , , and into the equation of the plane: Thus, the equation of the plane is: and the correct answer is (b).

Step 1: Understanding the geometry.
The intercepts of the plane with the axes form a relationship based on the geometry of the situation. By applying properties of planes and intercepts, we find the sum of the reciprocals equals .

Step 2: Conclusion.
The sum of the reciprocals is , corresponding to option (1).

The equation of the plane is derived by using the perpendicular condition with respect to the two given planes, and using the point .

We compute the shortest distance between two skew lines using the formula for the distance between two non-parallel lines.

For two lines to be perpendicular, the condition must hold, which follows from the geometric properties of line slopes.

Direction cosines of a line are calculated using the angles it makes with the coordinate axes. Here, the direction cosines corresponding to the given angles and are computed.

The center of the sphere is . The coordinates of the other end of the diameter are calculated as the midpoint of the center and the given point.

The shortest distance between two skew lines can be found using the formula involving the direction ratios and a vector joining the lines. After applying the formula, the shortest distance is found to be 1.
Step 2: Conclusion.
The correct answer is (C), 1.

To find the equation of the line passing through the origin and intersecting the given lines, use the concept of direction ratios and solve the system of equations by cross-multiplying the ratios.
Final Answer:

The distance from a point to a plane in the direction of a line can be computed by projection. First, express the line's direction ratios and then use the projection formula.
Final Answer:

Step 1: Identify the general equation of the line in the plane and use the geometric properties to find the value of .
Step 2: Substitute into the plane equation to solve for and determine the correct values.

Final Answer:

Step 1: The vector equation of a line is given as: where is a point on the line, and is the direction vector.
Step 2: Comparing the given equation with the standard form, the vector equation in the symmetrical form is:
Final Answer:

From the given vector equation, equate the components of the vector to find the value of . The solution leads to .
Final Answer:

The equation of the plane bisecting two other planes is found by averaging the coefficients of the given planes. The resulting equation is .
Final Answer: