In vector algebra I have learned that direction cosines of a vector specifies regarding the direction of a vector and that it is the equivalent of slope of a line in 2d. But when we derive the formula the formula of direction cosines we take a reference frame and form a position vector then we calculate the cosines of that position vector.
Now I have a huge confusion regarding if that is the cosines of a particular position vector. How can we use direction cosines as any parallel line which will will have the same direction cosines? Won't shifting the vectors positions change its coordinates and direction cosines? Then is that is so then why do we use just calculate the direction cosine of a vector and use it as a equivalent of all parallel vectors while calculating the l equation of line.
Maybe I could not explain my doubt but I want to know why on shifting the vector wrt to the reference frame does not change its direction cosines even if the coordinates of the head and tail of the vector change. Can anyone please help?please.
I think of a geometric vector as what some people call a "free" vector, as a set of ordered pairs of points, i.e. directed line segments, all in the same direction, so if $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_1,z_2)$ are points, we define the vector $\vec {P_1P_2}$ ,which we also write as $[x_2-x_1,y_2-y_1,z_2-z_1],$ as the set $$\{((x',y',z'),(x'',y'',z''))|x''-x'=x_2-x_1,y''-y'=y_2-y_1,z''-z'=z_2-z_1\}$$ so the vector$\vec {P_1P_2}$ is not just the directed line segment from $P_1$ to $P_2$, it is the set of all directed line segments in the same direction and having the same length as the directed line segment from $P_1$ to $P_2$. Then, if $(P_3,P_4)$ is any one of these directed line segments, $\vec {P_3P_4}=\vec {P_1P_2}$ i.e. a vector can be represented by any one of its member directed line segments. So to find a direction cosine, we just represent a vector by a directed line segment intersecting the coordinate axis involved-to find another of the direction cosines of the same vector we represent the vector by another (but note that it will be parallel to the first directed line sgment) directed line segment intersecting the coordinate axis we are considering now.