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Studying the basic concepts of vectors, I am very confused with the definitions of vector components and magnitudes. And why does the magnitudes always have to be positive? How about the components?

Thank you.

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A vector is a geometric object with both direction and "length."

Magnitude is basically the length of the vector from head to tail. The magnitude therefore has to be positive since lengths have to be positive.

A component is any individual entry in the vector. For example, the vector

$\vec{v}=\begin{pmatrix} 228 \\ 43 \\ 392 \\ 25 \end{pmatrix}$

is 4-dimensional and has components 228, 43, 392, and 25.

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A vector may be thought of as a directed line segment so that it has a magnitude (the size of the vector) and a direction. It is conventional (and almost common sense) to take the size of the vector to be a positive quantity. When you talk about the components you are using of the of vector you are asking for the size of that vector in some direction. For two dimensional vectors often (but not always) these directions you take are perpendicular to each other. In the attached figure the vector has components in the directions $x$ and $y$, the blue vector is that bit of the original vector that points along the $x$ direction and the red vector is that bit of the original vector that points in the $y$ direction.

enter image description here

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