# Projection of a vector, notation for

Consider an unit vector $$\vec d$$ (X coordinate) and another vector $$\vec v$$. All vectors are on a plane.

The X coordinate of $$\vec v$$ is given by the formula $$\vec{d}\cdot\vec{v}$$.

I need to write a formula for the Y coordinate of $$\vec v$$. I need Y coordinate approximately (up to $$O(|\vec v|^2)$$ big O notation).

The only formulas I know are: $$|\vec v|\widehat{dv}=|\vec v|\sin\widehat{dv}+O(|\vec v|^2)$$.

Is there an even simpler notation for this thing (widespread in the literature)?

Usually if you have bothered to define a unit vector in the $$x$$ direction, you have defined a unit $$y$$ vector as well. The unit vectors might be named $$e_1$$ and $$e_2,$$ or $$e_x$$ and $$e_y,$$ or $$\mathbf i$$ and $$\mathbf j,$$ depending on context.
So I would write $$e_1 \cdot \vec v$$ for the $$x$$ coordinate and $$e_2 \cdot \vec v$$ for the $$y$$ coordinate.
If you want to write the $$x$$ and $$y$$ coordinates of vectors for various purposes, you can identify $$e_1, e_2$$ (or some other notation) as the orthonormal basis of your space corresponding to the $$x$$ and $$y$$ coordinates, and then you can say things such as $$\vec v = (v_x, v_y)$$ over that basis. Then the $$x$$ coordinate of $$\vec v$$ is $$v_x$$ and the $$y$$ coordinate is $$v_y.$$ I don't know of a convenient "notation" to say "over that basis"; you just write it in words.
By the way, as an alternative to big-O notation, an exact (and more generally useful) equation involving the $$y$$ coordinate is $$(e_1 \cdot \vec v)^2 + (e_2 \cdot \vec v)^2 = \lVert \vec v\rVert^2.$$ This gives the result that the $$y$$ coordinate is $$\pm\sqrt{\lVert \vec v\rVert^2 + (e_1 \cdot \vec v)^2},$$ where you have to decide whether to use the positive or negative value depending on whether $$v$$ is on the "positive $$y$$" or "negative $$y$$" side of $$e_1.$$ Based on typical conventions, the sign of the $$y$$ coordinate would be the same as the sign of $$\widehat{dv}.$$