Why is it possible to get two new vectors that are perpendicular to each other from a single vector?

Answer 1

Because every vector is perpendicular to a plane and in a plane you can always find a vector perpendicular to any given vector.

Answer 2

I take your question to be more about how, rather than why it's possible. It's physically possible and mathematically possible (using trigonometry).
Physically: Consider a force F applied to the intersection of two rods x and y that that are hinged together and form a right angle, and suppose the rods are attached to stationary points via hinges. The rods can only transmit forces along their length, so the applied force will be resolved into two force components Fx and Fy perpendicular to each other whose vector sum = the applied force.
Mathematically: If you know the angle theta that F makes with one of the rods, you can predict Fx and Fy. Suppose theta is relative to rod x. Then Fx = F*cos(theta) and Fy = Fsin(theta).

Answer 3

it is possible because a number of vectors can be combining to form a single vector. those vectors that are combined to make a single vector are called components of the original vector. when you combine the components, the vector it forms is automatically perpendicular to each other because they intesect on a point.