Can the magnitude of a vector ever be equal - less then one of its components? Please explain,highly confused?

Question

Can the magnitude of a vector ever be equal - less then one of its components? Please explain,highly confused?

Answer 1

No, because the trig functions used to find components are never greater than 1. To break a vector into components, you are multiplying its magnitude by either the sine or the cosine of that magnitude. Since neither sine nor cosine can ever be greater than 1, the magnitude of the vector can never be less than one of its components.

Equality is possible, as a vector entirely plotted in one direction (say east) has an easterly component of ||v||*cos(0). Cosine of zero is 1 - thereby making the original vector equal in magnitude to its easterly component (also, that vector has no vertical component, as the sine of zero is zero).

Imagine the unit circle, and the vector starts out in the positive x-direction. The coordinates of this are then (0,1). Rotating it 90 degrees counterclockwise gives you (1,0). This process continues throughout the 360 degree revolution. Components are the resolving of the vector to a combination of these 'directions'.