Final answer:
To find the point closest to C on segment AB, find the equation of the line AB and calculate the shortest distance between C and the line. For the area of triangle ABC, use Heron's formula. To find the length of the altitude drawn from A to BC, find the equation of the line BC and calculate the shortest distance between A and BC.
Step-by-step explanation:
To find the point closest to C on segment AB, we first find the equation of the line AB. The line AB can be represented by the vector equation r = A + t(B - A), where A and B are the given points. Substitute the x, y, and z coordinates of A and B into the equation to find the equation of the line AB. Next, we find the shortest distance between the point C and the line AB. This can be done by finding the perpendicular distance from C to the line AB. Use the formula d = |(C - A) · ((B - A) x (C - A))| / |B - A|, where · represents the dot product and x represents the cross product. Substitute the coordinates of A, B, and C into the formula and calculate the distance to find the point closest to C on segment AB.
To find the area of triangle ABC, we can use Heron's formula. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by A = √(s(s - a)(s - b)(s - c)), where s is the semi-perimeter of the triangle. Calculate the lengths of the sides AB, BC, and CA using the distance formula, and substitute them into Heron's formula to find the area of triangle ABC.
To find the length of the altitude drawn from A to BC, first find the equation of the line BC. The line BC can be represented by the vector equation r = B + u(C - B), where B and C are the given points. Substitute the x, y, and z coordinates of B and C into the equation to find the equation of the line BC. Next, find the intersection point of the line BC and the line perpendicular to BC passing through A.
The shortest distance between A and BC is the length of the altitude. Use the formula d = |(A - B) · ((C - B) x (A - B))| / |C - B|, where · represents the dot product and x represents the cross product. Substitute the coordinates of A, B, and C into the formula and calculate the distance to find the length of the altitude drawn from A to BC.