Since momentum is the product of a vector and a scalar, momentum is also a vector

Explanation

That is true.

To see what effect this has, let's see this example of the pool table.  In part 1 of the figure, ball A is about to collide with ball B, but it is coming in a direction such that it will not hit straight on.  You would expect the balls to continue somewhat as shown in part 2.  Before the collision, the total momentum of the system was in a downward direction on the page.  After the collision, the momentum of ball A is not only downward, but also toward the right.