Why Do we call a projection of one vector onto another a "component"? I realize the component of one vector $\vec{a}$ in the direction of the other vector $\vec{b}$ is essentially a projection of $\vec{a}$ onto $\vec{b}$ and I understand how it is calculated. What I don't understand is why exactly we call this projection a "component" in this case. If we take vector $\vec{a}\space(5, 6)$ in $2D$ space with origin in $(0, 0)$, what I am used to call the components of such vector are vectors $\vec{b}\space(5, 0)$, and $\vec{c}\space(0, 6)$. Or, in simple words, such vector has $5$ units towards $X$ direction, and $6$ untis towards $Y$ direction.
Now, looking at this picture projection of $\vec{a}$ onto $\vec{b}$ we still call the resulting projection of $\vec{a}$ onto $\vec{b}\space$ a "component". However this component does not have anything to do with components of vector $\vec{a}$ that are along $X$ or $Y$ axis. I would need some intuition on why we call the resulting projection of one vector onto another a component. It's more intuitive for me to just call it a "projection", not a "component".
 A: The word component here for a vector $\mathbf{v}$ means that $\mathbf{v}$ is already known as a (linear) composition of vectors, that is, that in the given context we have that $\mathbf{v}=\mathbf{v}_1+\ldots +\mathbf{v}_n$ for some vectors $\mathbf{v}_1,\ldots ,\mathbf{v}_n$. Then we can say that each $\mathbf{v}_k$ is a component of (the given/known composition of) $\mathbf{v}$.
Also we can say that $\mathbf{v}$ can be decomposed as the sum of a list of vectors $\mathbf{v}_1,\ldots ,\mathbf{v}_n$, this just means that $\mathbf{v}=\mathbf{v}_1+\ldots +\mathbf{v}_n$, then in this context a component (of the given decomposition) is (again) just one the $\mathbf{v}_k$ vectors already mentioned.
Then we can also talk of a decomposition of $\mathbf{v}$ in a list of linearly independent, or orthogonal, vectors. The kind of decomposition used will depend on the context and why we are decomposing $\mathbf{v}$ in the given way.
A: Because every vector is defined with respect to a coordinate system. If your vector $b$ is the unit-vector $(1,0)$ your are indeed computing the "component" in the X-axis. If $b$ would be the vector $(0,1)$ you would compute the component in the Y-axis. But no one forces you to use the X- or Y-axes. You might use another basis to generate your vector $v$ in the $2$-D plane. If you use the standard X,Y-axes then you would you as basis $B=\{(0,1), (1,0)\}$. But you could simply project in any other vector and define it as the component on such vector.
