Problem in Vector Resolution and Component

[avistein]

I cannot understand what is vector resolution.It is said in the book that ON is the resolved part of A along X axis.It is also known as the x-component of A or the horizontal component of A.Ax may be regarded as the projection of A on X-axis. OM is the the resolved part of A along Y-axis.It is also known as the y-component of A or vertical component of A.The vertical component of A may be regarded as the projection of A on Y-axis.Now what is that projection? Is it the image of A on X-axis or Y axis?
Then why Ax+Ay=A and not A=Ax or A=Ay?If Ax and Ay are the images of A on X and Y resp. then the magnitude of Ax and Ay should be same as A,but no, why? Please explain me.I am very much confused.

 

[mathman]

If A = (a,b), then the x-axis projection is (a,0) while the y-axis projection is (0,b).

It is quite simple - don't let the terminology confuse you.

 

[deepani]

vectors are simple

Now what is that projection? Is it the image of A on X-axis or Y axis?
Then why Ax+Ay=A and not A=Ax or A=Ay?If Ax and Ay are the images of A on X and Y resp. then the magnitude of Ax and Ay should be same as A,but no, why?

Don't get muddled up. Let, me explain what projection is. Say, the vector extends from (0,0) to (a.b). Suppose, you want projection on the x-axis, take a light source and place it directly above the end of the vector, the shadow would be at (a,0). Thus the projection of the vector extends from (0,0) to (a,0). similarly, y-component would extend from (0,0) to (0,b).
And, by the way A_x+A_y ≠ A. Using the Pythagoras theorem, (A_x)²+(A_y)² = A².
What the book might have meant would have been, was vector A_x+vector A_y = vector A. By writing vector, I am also considering the direction. While, above, I was only talking about magnitudes. With a little practice, you would easily understand the difference between the vector and it's magnitude. So, good luck!