What does it mean if a vector name has an → above its name

Question

I am doing a course and one of the mathematical algorithms which were mentioned included the name of a column vector (a m*1 Matrix) with an arrow (→) above its name. What does it mean?

Answer 1

A vector defined as $\vec{v}$ has a specific meaning. The arrow on the top denotes a specific orientation in the space. Let us say that a vector $\vec{v}$ lies in our 3D real space $\mathbf{R}^3$. Then, geometrically the $\vec{v}$ could mean a directed line segment from the origin (taken to be (x=0,y=0,z=0)). The components of the vector in the respective axes i.e. the x-, y- and z- axis give the location coordinates of the tip of the vector, while the tail rests at the origin. For example, lets define an arbitrary vector in $\mathbf{R}^3$ as $$\vec{A}=a_x \hat{i}+ a_y \hat{j} +a_z \hat{k}$$

Here, the triad $\{a_x,a_y,a_z\}$ are the coordinates specifying the amount that each component extends along the respective axes i.e. how many units of distance to move. The $\{\hat{i},\hat{j},\hat{k}\}$ are the unit vectors that indicate the direction along which to move.

In linear algebra, however, the vectors are not thought of as directed line segments and hence, they do away with the need of arrows over their heads. Instead they use the bold notation as $\mathbf{v}$ to define a vector.

In 3D, such a vector would be written simply as $$\mathbf{v}= [a_x,a_y,a_z]$$

in terms of just its components. Of-course the components will change depending upon which basis you choose to define your vector i.e. Cartesian basis like $\{\hat{i},\hat{j},\hat{k}\}$, cylindrical coordinate basis $\{\hat{\rho},\hat{\phi},\hat{z}\}$ or spherical coordinate basis $\{\hat{r},\hat{\theta},\hat{\phi}\}$.