Question

Explain why a vector cannot have a component greater than its own magnitude

Answer 1

No,component of vector cannot have magnitude greater than the vector .

Lets say A is vector in a two dimensional x-y plane then projection on x axis that is x component is equal to Ax=|A|cos and Ay=|A|sin . Since value of cos and sin is between  -1<x<1. So magnitude of component is always lesser than vector.

geometric explanation .

let Ax, Ay be the horizontal and vertical components.
The reason that it can't be greater is that when you join the components from head to tail, it forms a right angled triangle, and the hypotenuse that is the formed by vector is always greater than its sides.