Final answer:
To find the plane through three given points, use the cross product to find a normal vector and then the plane equation. The line perpendicular to the plane through a point has that normal vector as its direction. Use the dot product and cosine formula to find the angle between two planes, and the cross product of their normals for the intersection line's direction.
Step-by-step explanation:
To find the equation of the plane passing through points A(2,1,1), B(-1,-1,10), and C(1,3,-4), we can use the vector cross product to find a normal vector to the plane. Let's call the vectors formed by these points πBA and πCA. The normal vector (n) of the plane can be found by the cross product of πBA and πCA. Once n is found, we can write the equation of the plane using point A and the normal vector as Ax + By + Cz = D.
For the line through point B perpendicular to the plane, the direction vector will be the same as the normal vector of the plane. The symmetric equations of the line can be expressed based on B's coordinates and the normal vector.
To find the angle between the planes, we use the dot product of their normal vectors and apply the cosine formula. We have one normal vector from the first plane and we're given the second normal vector. By taking the inverse cosine of the dot product over the magnitudes of the normal vectors, we can find the acute angle between the planes.
Finally, for the line of intersection, we look for a point that lies on both planes, and we can use the cross product of their normal vectors to find the direction vector of the intersection line. Using this point and direction vector, we can express the parametric equations of the line of intersection.