vectors
I am quite confused about vectors and vector quantity. Somewhere I find they are used interchangeably and somewhere, they have different meanings. So, I wish to know whether there is a difference between vector and vector quantities or not. Is force a vector or a vector quantity. Is vector a directed line segment or a physical quantity which has both magnitude and direction and add according to vector laws of addition?
Please help me.
Elements of a real vectorspace certainly have direction, but they don't really have a magnitude. Well actually, they... kind-of have a magnitude. But for a proper magnitude, you need further structure, such as a norm or inner product. Let me explain.
Vector Spaces.
Suppose $V$ is a real vectorspace.
Definition 0. Given a vectors $x,y \in V$, we say that $x$ and $y$ have the same direction iff:
- there exists $r \in \mathbb{R}_{\geq 0}$ such that $x = ry,$ and
- there exists $r \in \mathbb{R}_{\geq 0}$ such that $y = rx$.
(The $r$'s don't have to be the same.)
This induces an equivalence relation on $V$, so we get a partitioning of $V$ into cells. Each cell is an open ray, so long as we regard $\{0\}$ as an open ray. You may wish to exclude $\{0\}$ from its privileged position as a ray, in which case you should only deal with non-zero vectors; that is, you need to be dealing with $V \setminus \{0\}$ rather than $V$.
Irrespective of which conventions are used, we can make sense of direction using these ideas:
Definition 1. The direction of $x \in V$ is the unique open ray $R \subseteq V$ such that $x \in R$.
Notice that the equivalence relation of having the same direction is preserved under scalar multiplication; what I mean is that if $v$ and $w$ have the same direction, then $av$ and $aw$ have the same direction, for any $a \in \mathbb{R}$. Geometrically, this means that if we scale a ray, we'll end up with a subset of another ray.
As for magnitude; well, if you choose a ray $R \subseteq V$, then we can partially order $R$ as follows. Given $x,y \in R$, we define that $x \geq y$ iff $x = ry$ for some $r \in \mathbb{R}_{\geq 1}$. So some vectors along this ray are longer than others, hence magnitude.
Inner Product Spaces.
Actually, this isn't the whole story. The problem with vector spaces is that if $x$ and $y$ don't belong to the same ray (nor to the the "negatives" of each others rays), then there's no way of comparing the magnitudes of $x$ and $y$. We can't say which is longer! Now there are mathematical situations where this limitation is desirable, but physically, you probably don't want this. A related issue is that you can't really make sense of angles in a (mere) vector space; at least, not without some further structure.
For this reason, when physicists say "vector", what they usually mean is "element of a finite-dimensional inner-product space." This is a (finite-dimensional) vector space $V$ with further structure; in particular, it comes equipped with a function
$$\langle-,-\rangle : V \times V \rightarrow \mathbb{R}$$
that is required to satisfy certain axioms resembling the dot product. Especially important for us is that these axioms include a "non-negativity" condition:
$$\langle x,x\rangle \geq 0$$
Using this, we can define the magnitude of vectors as follows.
Definition 2. Suppose $V$ is a real inner product space. Then the norm (or "magnitude") of $x \in V$, denoted $\|x\|$, is defined a follows:
$$\|x\| = \langle x,x\rangle^{1/2}$$
This allows us to compare the magnitudes of vectors that don't live in the same ray; we simply define that $x \geq y$ means $\|x\| \geq \|y\|.$ When confined to a single ray, this agrees with our earlier definition! Be careful though, because the relation $\geq$ we just defined is only a preorder.
In fact, the inner product gives us more than just magnitudes; it also gives angles!
Definition 3. Suppose $V$ is a real inner product space. Then the angle between of $x,y \in V$, denoted $\mathrm{ang}(x,y)$, is defined a follows:
$$\mathrm{ang}(x,y) = \cos^{-1}\left(\frac{\langle x,y\rangle}{\|x\|\|y\|}\right)$$
It can be shown that vectors $x$ and $y$ have the same direction (in the sense described at the beginning of my post) iff the angle between them is $0$. In fact, you can modify the above definition so that it defines the angle between any two non-zero open rays. In this case, it turns out that two rays are equal iff the angle between them is $0$.
A single force can be represented by an arrow (a simple assumption)
An arrow is one-dimensional.
As by Google, a
Vector is:
a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.
Quantity is:
the amount or number of a material or abstract thing not usually estimated by spatial measurement.
We know that the length of an arrow represents the power of a force, or in other words, the quantity. Also, we know that the direction of an arrow is it's, well direction, and the magnitude being the quantity. And, also, it is actually determining the position of a point in space relative to another! The comparison is, well an origin on a set of coordinates, longitude and latitude in real life, and there are also many more examples!
From this, we conclude that it is not necessary to classify as something else.
As for your last question, I don't understand what is meant by such quantities, so therefore, I cannot answer that (probably because of my poor comprehension...)
Best Answer
Saying that force is a "vector quantity" is usually just a way of saying that force is always represented by a vector. If we were to say that force is a "vector", the implication would be that force was only allowed to ever be one vector. However, no one would look sideways at you if you said that force was a vector; it's a very pedantic distinction.
A vector is a mathematical object which we can work with in the following ways:
This definition then allows you to model a lot of interesting physical quantities, like force. In particular, a particular class of vectors, which we denote by $\mathbb{R}^2$, is represented by arrows in the plane which we add and multiply by scalars in the way you're familar with. It's this set of vectors which we usually use to describe force. Force is a vector quantity, meaning that it is always described by a vector $v$, and the laws pertaining to force always involve vector arithmetic for this reason.