Point multiplication in elliptic curve

Question

Suppose $a$ is an integer and $Q$ is a point on an elliptic curve and $(x,y)$ are $x$ and $y$ coordinates of this point.
My question is:
Whether $a\cdot Q$ is equal to $(ax, ay)$?

Answer 1

The answer is no. See how the addition on a elliptic curve is motivated by geometry: Adding $P$ and $Q$ is done by finding the third point of intersection between the line through $P$ and $Q$ and the curve, then reflecting at the $x$-axis. For $P+P$ you take the tangent at $P$, find the second intersection point and reflect that. This will in general not coincide with just adding the coordinates.

This is illustrated in the Wikipedia article on Elliptic Curves.