What exactly does it mean for a vector to have a direction?

Question

Vectors are defined as having magnitude and direction. If I understand it correctly, their magnitude is their length, meaning they have the properties of a line segment. What does it mean for a vector to have a direction? Let me be more specific:

Let $\vec v=\begin{bmatrix}2\\3\end{bmatrix}$

The angle of this vector is $\arctan(\frac y x)=\arctan(\frac3 2)$ from the positive $x$ axis and we know that the direction of this vector is "pointing to the upper right". What does this mean geometrically? What defines direction for a vector? (I am new to the topic so go easy on me.)

Answer 1

That is the correct definition in physics. If you incorporate the correct definition of a vector in maths, you can easily understand the "direction" concept alternatively.

In physics the direction associated with a vector means you first fix a coordinate system and see in which direction the effect of the vector is as a whole. For example take velocity v.

So say $v=a\hat{i}$. This means that the velocity is entirely in the direction of x axis of your chosen coordinate system. That effect of a vector is 0 in the direction perpendicular to the direction in which it's effect is as a whole. Now you could represent the same vector for velocity in another basis. But logically changing basis should not change the vector ( the velocity of any object has to be in the same direction in which ever coordinate system you represent it). So in the new coordinate system the entire effect of the vector ( v here) would still be in the x direction ( of your previous coordinate system) with the only change that you are representing your vector in a different coordinate system.

Mathematically a vector is an element of a vector space and there are vectors with which you cannot associate any sense direction. Like a wavefunction is a vector but you cannot define a direction with it. A matrix is a vector but you cannot associate a direction with it in the normal sense.

I should mention that only in our 3 dimensional space or Euclidean space can we sensibly denote a sense of direction to a vector and not generally in other spaces. Because our sense of direction itself doesn't exist for other spaces. You will need to first define what a direction means in a vector space and only then can a vector in that space be given a sense of direction. That direction might be completely different in sense than the directions we are used to understand in our world. So you see how algebra helps us do things in higher dimensions where geometry couldn't.

I hope that helped

Answer 2

It's a very good question! It seems basic at first glance but holds a little more complex nuance.

When you write $v^\to =[23]$ you basically write that, according to the standard coordinates (that is cartesian ones) $v$ goes 2 increments along the $x$ axis and 3 along $y$. however that depends on the system used (meaning the base); When you change bases you change the coordinate system and therefore $v$'s identification will change.

Answer 3

The direction of a vector can be formalized in a couple of different ways.

In Euclidean space, one can define a direction vector to be a vector $\vec u$ such that $|\vec u| = 1$. So, for example, the direction vectors in the plane are those vectors based at the origin whose tip is on the unit circle, and therefore the direction vectors are in one-to-one correspondence with angles chosen in the interval $[0,2\pi)$, where the direction vector corresponding to the angle $\theta \in [0,2\pi)$ is $\vec u = \langle \cos \theta, \sin \theta \rangle$. Next, given an arbitrary nonzero vector $\vec v \ne \vec 0$ in Euclidean space, one can define the direction of $\vec v$ to be the direction vector $$\vec u = \frac{1}{|\vec v|} \vec v = \biggl\langle \frac{v_1}{\sqrt{v_1^2+\cdots+v_n^2}},..., \frac{v_n}{\sqrt{v_1^2+\cdots+v_n^2}} \biggr\rangle $$ And now it's clear what it means to say that two vectors $\vec v$ and $\vec w$ have the same direction: it means that $\frac{1}{|\vec v|} \vec v = \frac{1}{|\vec w|} \vec w$.

Another way to formalize direction, which works in any vector space $V$ regardless of whether it has a norm, goes like this. Consider the set of nonzero vectors $V - \{\vec 0\}$. Define an equivalence relation on $V - \{\vec 0\}$, where $\vec u, \vec v \in V-\{\vec 0\}$ are equivalent if there exists a scalar $r > 0$ such that $r \vec u = \vec v$. One can formally define the direction of a nonzero vector to be its equivalence class, under this equivalence relation. So to say that two nonzero vectors $\vec v,\vec w$ have the same direction means that they are equivalent. In fact you can even formally define the direction of $\vec v$ to be its equivalence class, namely the subset of all vectors $\{r \vec v \mid r > 0\}$; geometrically, this is just the open ray parallel to $\vec v$.