Shortest Distance Between Skew Lines

Let the coordinates of point be . The distance from to is: The distance from to is: We are given that the distance from to is twice the distance from to , so: Squaring both sides: Expanding both sides: Simplifying the equation: Dividing by -3: This is the equation of the locus of point . Step 2: The equation of the line is: Substitute this into the equation of the locus of : Expanding the square term: Combining like terms: Multiplying through by 9 to eliminate fractions: Divide by 8: Now solve this quadratic equation using the quadratic formula: Thus, the two values of are: The distance between and is: Thus, the distance between and is .

We are given the vertices , , , and of a tetrahedron, and we are asked to find the cosine of the angle between the faces and .
Step 1: The angle between two planes is given by the formula: where and are the normal vectors to the planes. For the face , the normal vector is the cross product of vectors and , and for the face , the normal vector is the cross product of vectors and .
Step 2: First, find the vectors:
Step 3: Now, compute the cross products: For : Thus, . For : Thus, .
Step 4: Now, calculate the dot product : Next, find the magnitudes of and : Step 5: Now, calculate : Thus, the value of is .

Step 1: Identify the center and radius of the circle. The equation can be rewritten by completing the square: Thus, the circle has center and radius .
Step 2: Calculate the distance from point to the center of the circle: Step 3: The largest distance from to any point on the circle is the sum of the radius of the circle and the distance from to the center of the circle. Therefore: However, the correct final answer to this question should be based on the distance interpretation provided.