Computational geometry

1. What is the difference between a point and a vector?
2. What is the dot product of two vectors? List its properties.
3. How do you check if two vectors are parallel/orthogonal?
4. How do you find a projection of one vector onto the direction of another vector?
5. What is the cross product? List its properties.
6. How do you compute an angle between two vectors?
7. How do you compute a signed angle measured from vector A to vector B on a 3D plane containing both vectors?
8. How to check if three vectors are coplanar?
9. Can you uniquely define a plane having two non-collinear vectors? How?
10. What is the mixed product of three vectors? What is its geometric sense?
11. What is a convex polygon? How do you check if a polygon is convex or concave?

General questions
1. Compute the distance between an infinite line and a point in 3D. What needs to be changed to compute the distance between a finite segment and a point?
2. Compute the distance between two arbitrary infinite lines in 3D.
3. Compute the intersection line for two planes.
4. How do you compute the area of a convex polygon? How should the algorithm be changed to work with non-convex polygons?
5. How do you determine whether a polygon's points go in a clockwise or counter-clockwise direction?
6. How many parameters are required to uniquely define a line/plane? What are the parameters?
7. How many parameters are required to uniquely define a sphere/cone/cylinder/torus? What are the parameters?
8. Is it possible to find a circle's radius and the center point from three points lying on the circle? If yes, how would you do so? Is it possible to do with less than 3 points? 
9. Suppose you have a triangle. All edge lengths and corner angles are known. How would you compute the radius of the minimum enclosing circle?

Transformations and rotational matrices
1. Given a coordinate system with X-axis (x1, y1), Y-axis (x2, y2) and an origin O(x,y). What would be the matrix that transforms the point coordinates from a global coordinate space?
2. How do you turn a 2D vector 90 degrees in a CCW direction?
3. How do you compute an inverted rotational matrix? 
4. What would be the transformation matrix that rotates an object around an axis that passes through its origin and is aligned with the Z-direction? How would the matrix change if the axis passed through an arbitrary point P in 3D space?
5. What would be the matrix for a 0 degree rotation around an axis line that passes point P(0.55, 12.669, 0.15) and has the direction D = (‑0.002196, ‑0.0020014, ‑0.9999956)?
6. Given two coordinate systems CS1 and CS2 defined by transformation matrices M1 and M2 correspondingly. Given a point P in CS1's space. How do you compute the coordinates of P in CS2's space?

Literature