Hint for 1:

the vector parallel to the vector a is a constant times the vector a. a vector perpendicular to a is one for which its dot product with a is zero

you want these two vectors to add and give you b, you will have to solve for some unknowns

Hint for 2: if the plane is parallel, it has the same normal vector, namely <1,3,-5>.

a plane with normal vector passing through a point is given by

the distance, , from a point to a plane is given by:

Hint for 3: find c, let and take its dot product with b and find the vector when you subtract it from a, set the two vectors equal and solve for the components

Hint for 4: do the computation and set it equal zero. you should be able to see if this describes a line or not.

then set the same computation to a nonzero constant, show that you can't get a line

Hint for 5: come on, finding determinants is something you can look up, quite easily. google is your friend.

Hint for 6: again, something you can look up.