Problem 3: You can prove the identity by brute force. Use the coordinates of the vectors and plug them into the left hand side of the identity. You will see how everything cancels. However, by distributivity of the dot product, you can get an one-line proof. Keep in mind that the dot product of two vectors is no longer a vector, but a scale.
Problem 5: Students are supposed to see how this problem is related to the projection and the orthogonal projection of a vector.
Problem 9: You should choose one of those points as the initial point of the vectors and other three as terminal points of the vectors. Then apply the triple product to those three vectors to see if it is zero.
Problem 10: Strictly speaking, the vectors should be half of the cross product of certain vectors. Most of the students forgot to put the half in their expressions. A lot of students put bars around the cross product. This is wrong because we are looking for the vectors representing each face of the tetrahedron but not the magnitude of them. Using the right hand rule to decide the direction of the vectors is also crucial here.