3d Geometry

Direction vector of line = Normal vector of plane = If dot product = 0 → perpendicular. Wait! But direction vector of line is same as normal to plane ⇒ they are perpendicular.

Parametrize line with parameter : Substitute into surface: Solve this equation to find values of , then plug back into line equations to get coordinates. Final points are: (3, 4, -2) and (6, -2, 2).

Given, , and the line makes equal angles with the X and Z axes.
This implies that for the line will satisfy the equation: Substitute the value of and solve: Since , we substitute into the equation, and solving the equation gives .

The distance from a point to a line defined parametrically as is given by the formula: Substituting the values:

We are asked to find the point on the sphere closest to the plane.
This can be done using the distance formula between a point and a plane.
By solving the equation, we find that the correct point is .

We are given the coordinates where . The total number of points in the set is . An equilateral triangle is formed by selecting any 3 points that are equidistant from each other. The number of such triangles that can be formed from the set of 27 points is 72.

The formula for the length of the angle bisector is: where , , and are the sides of the triangle. After computing the side lengths and substituting into the formula, we find that the length of the bisector is .