Why are components of vectors scalars?

Question

Lets say we have a vector 1i+2j, I saw that the components of the vectors are said to be scalars, how can that be possible, doesnt 1i have a direction, also if they were a scalar why are the following vector laws of addition,

Answer 1

It's probably inviting down-votes to quote a Physics textbook on the Mathematics Stack Exchange, but at least this book, Principles of Mechanics by Synge and Griffiths, was well-respected in its time. The writers recognise that some people use "components of vector, V" to mean vectors in chosen directions that add (vectorially!) to make V. S and G call these, "vector components of V". These vector components can be thought of as products of unit vectors in the chosen directions, and scalar coefficients. Synge and Griffiths call the coefficients "the scalar components of V".

Distinguishing between "vector components" and "scalar components" removes ambiguity in how the term is being used. Although S and G were dealing exclusively with vectors in 3-dimensional space, the distinction could surely be carried over to more abstract vectors.

Answer 2

The components in this context aren't $1\mathbf i$ and $2\mathbf j$, but $1$ and $2$.

And indeed, $\mathbf i$ and $\mathbf j$ are vectors, which can be (and are) scaled to make linear combinations.

Answer 3

1. The components of $1 \hat{i}+2\hat{j}$ are $1,2$ and not $1 \hat{i}, 2\hat{j}$

2. Components of vectors are not scalars, and not vectors- they are simply components of vectors. Scalars are objects who do not change under change of coordinates (e.g. if you rotate the axes by some angle). The components of a vector do change. Example: if we take some cartesian coordinates $x,y$, and the vector $\vec{V}= 2 \hat{x}+5\hat{y}$, if we take $x',y'$ to be coordinates which are rotated by $\pi/2$ we get $\vec{V}= 5 \hat{x'}-2\hat{y'}$. The components changed. The vector length, hoever, is a scalar, as you can check.