Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be i^+2j^+k^, i^+3j^−2k^ and 2i^+j^−k^ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is 3110 and the volume of the tetrahedron is 62805, then the position vector of E is
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Step-by-Step Solution
Step 1: Calculate vectors AB and AC and area of triangle ABC
First, we find the vectors AB and AC by subtracting the position vector of A from B and C respectively. Then, we calculate the cross product of AB and AC to find a vector normal to the plane containing triangle ABC. The magnitude of this cross product, divided by 2, gives the area of triangle ABC.
Step 2: Find the height of the tetrahedron
The volume of a tetrahedron is given by the formula 31×base area×height. Using the given volume and the calculated area of triangle ABC, we can find the height h of the tetrahedron from vertex D to the base ABC. The height is the projection of AD onto the normal vector of the plane ABC.
Step 3: Determine the position vector of D
We are given the length of AD. We can set up equations using the dot product of AD with the unit normal vector to the plane ABC, which equals the height h. We also use the magnitude of AD. Solving these equations will give us the coordinates of D. For simplicity, we will assume one of the possible solutions for d.
Step 4: Find the midpoint M of BC and the equation of the median AM
The median line segment through A connects vertex A to the midpoint M of the opposite side BC. We find the position vector of M by averaging the position vectors of B and C. Then, we find the vector AM and write the parametric equation of the line AM.
Step 5: Find the point E
The point E is the intersection of the altitude from D and the median AM. The altitude from D is a line passing through D and parallel to the normal vector of the plane ABC. We set up the parametric equations for both lines and equate their components to find the values of the parameters s and t. Substituting these values back into either line equation gives the position vector of E.