DIfference between vectors and relations of points

[Mr Davis 97]

When it comes to analytic geometry, I am little confused about the use of vectors. For example, throughout high school, one works in $\mathbb{R}^2$, and geometric objects such as lines are described using equations relating two variables, the x and y coordinate, such as y = 2x + 1. However, when we move to three-dimensions, it seems that vectors as well as relations between coordinates are used. For example, to represent a line with vectors, we might use something like $\vec{r} = \vec{r}_0 + t \vec{v}$, while we can also represent the same line with the simultaneous equation such as $x = 2y + 1,~ y = 3z - 5$, just as a random example. And for planes , we just use $ax+by+cz+d=0$. This leaves me wondering why we use vectors at all. It seems like vectors are another a way just to represent a triple of coordinates. If this is the case, why don't we just use equations relating variables all the time?

 

[Stephen Tashi]

Most people (who master vectors) find that it's easier to solve problems using vectors than using analytic geometry. For example, I suppose finding the (smallest) angle a given line makes with a given plane can be done with analytic geometry. However, most people find it easier to conceptualize the problem as projecting a vector that points along the line onto the plane and finding the angle between the vector and its projection. Can you conceptualize that problem just by thinking in terms of analytic geometry? Some people probably can, but I sure find it easier to think about in terms of vectors.

 

[Mr Davis 97]

What actually makes vectors unique though? Don't we represent vectors as a triple of coordinates, just as we would represent a point in analytic geometry as a triple of coordinates? Also, why aren't students taught about vectors first in their use to represent geometric objects as opposed to learning the analytic method first?

 

[ehild]

Vectors in the three-dimensional space are "real" things, having magnitude and directions. They can be represented in different ways: with three Cartesian coordinates or with a magnitude and two angles, but drawing a straight line segment of magnitude proportional to that of the vector, and direction, parallel to the vector, is also possible. You can add vectors geometrically, or using the coordinates.
In my country, students learn about vectors quite early, and learn how to add and subtract them, and about scalar product of vectors well before they start analytic geometry.
The meaning of vector is more general then the vectors in three dimensional space around as, connected to positions. Velocity and acceleration are also vectors. Electric and magnetic fields are also vector quantities.

 

[Mark44]

For example, to represent a line with vectors, we might use something like $\vec{r} = \vec{r_0} + t \vec{v}$, while we can also represent the same line with the simultaneous equation such as $x = 2y + 1,~ y = 3z - 5$, just as a random example.

The vector form you showed above is lots more intuitive to work with than your example with the two equations. Using the vector parametric form, draw a vector from the origin in the direction of $\vec{v_0}$. Its magnitude gets you to a point on the line. To get any other point on the line, choose a value of the parameter t, and extend a vector along the line to the point in question.

Here's a picture to help get the idea across. I did this in Paint, which doesn't have many capabilities for nice text. By vector addition, you should be able to see that $\vec{v_0} + t\vec{v} = \vec{r(t)}$.

The two equations in your example represent two planes that intersect in a line. Think about the effort you would need to expend to get a single point. Even so, you wouldn't have any idea about the direction of the line.

 

[Mr Davis 97]

So is the gist that for 2D geometry in the coordinate plane, using relations between variables suffices because it is rather simple. But once we go to higher dimensions, like 3D (or maybe even 4D), it is much easier to think in terms of vectors?

 

[mfb]

Right.

Even for two dimensions, vectors can be easier, especially if you want to calculate angles between something,

 

[Stephen Tashi]

What actually makes vectors unique though? Don't we represent vectors as a triple of coordinates, just as we would represent a point in analytic geometry as a triple of coordinates?

We don't take an inner product between two points. We don't subtract one point from another to obtain another point.

Also, why aren't students taught about vectors first in their use to represent geometric objects as opposed to learning the analytic method first?

I'm sure it's partly due to tradition.

However, I wouldn't say that students (in the USA) learn to represent geometric objects using analytic geometry. If you look at old time texts on analytic geometry (e.g. a text for that would be used in a 1-semester course), you will see that students (in the USA) are not taught much of that material.

It is appropriate to teach some topics from the viewpoint of coordinates - just to teach coordinates. There are phenomena that have coordinates, which don't represent points in Euclidean space (e.g. Data consisting of n-tuples of numbers that do not represent a geometric object - e.g. (height, weight, monthy income, ...) .) There are data that don't obey the parallelogram law, so they aren't vectors.

A function consists of data (x,y) and often the variables x and y are from the same "dimension". The graph of a function like y = x^2 can be regarded as a geometric object in 2-D, but it isn't a geometric object in the same sense that a square or triangle is a geometric object.

 

[Mr Davis 97]

Another question: If Euclidean vectors and coordinate vectors are distinct objects (the former has a notion of magnitude and direction while the latter does not), why do we often call both spaces $\mathbb{R}^n$? This would seem to imply that they are just the same type of object, that is, tuples of scalars.

 

[Stephen Tashi]

Another question: If Euclidean vectors and coordinate vectors are distinct objects (the former has a notion of magnitude and direction while the latter does not), why do we often call both spaces $\mathbb{R}^n$? This would seem to imply that they are just the same type of object, that is, tuples of scalars.

The ambiguous use of "$\mathbb{R}_n$" to denote both a "space" and a "vector space" is another example of customary mathematical transgressions in notation.

For example, the polar notation for a vector in 2-D produces a coordinate that is an element of the "space" $\mathbb{R}_2$. However, if someone speaks of the "vector space $\mathbb{R}_2$" they refer to the vector space whose operation on 2-tuples is the one appropriate for coordinatizing a vector using cartesian coordinates.

 

[Mark44]

Another question: If Euclidean vectors and coordinate vectors are distinct objects (the former has a notion of magnitude and direction while the latter does not), why do we often call both spaces $\mathbb{R}^n$? This would seem to imply that they are just the same type of object, that is, tuples of scalars.

What you're calling a "coordinate vector" is an ordered n-tuple that does have a magnitude, but may or may not have a direction. For example, the vector <2, 3, 0, -5, 7> is a vector in $\mathbb{R^5}$ but one would be hard pressed to say which direction it points. The magnitude of this vector is $\sqrt{2^2 + 3^2 +0^2 + (-5)^2 + 7^2} = \sqrt{87}$, using the usual Pythagorean measure.