When t = 0, F is just the identity map on R 2 . But as t grows from 0 to 1, each vector in R 2 beginning at the origin is stretched in length until the end of the isotopy, when they are all twice as long as they originally were. what does origin mean

In this context, origin means the point (0,0)(0,0)(0,0) in R2\mathbb{R}^2R2, where the x-axis and y-axis cross. It is the fixed starting point from which vectors are measured.

So when the sentence says “each vector in R2\mathbb{R}^2R2 beginning at the origin,” it means each vector starts at that zero point and points outward from there.

A simple way to picture it:

• The origin is the home base of the coordinate plane.
• A vector “beginning at the origin” has its tail at (0,0)(0,0)(0,0).
• Its length can then be stretched while keeping the same direction.

Would you like a quick sketch-style explanation of the whole sentence?