Thread: Vector

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

A vector is a quantity that has both magnitude and direction... such as your example, velocity. This is different from 'scalar' quantities, such as temperature, which have magnitude, but no direction. you can also look at them more abstractly as just being a certain collection of scalars.

For a velocity vector, V=<v_x,v_y>, you can say:

- It has 2 components, v_x and v_y...(It is 2-dimensional)
- v_x is the velocity in the x-direction
- v_y is the velocity in the y-direction.
- It has a direction θ (angle from the x-axis)
- It has a magnitude, |V|, which is its length. A right hand triangle will be formed with the vector, and it's components, where |V| is the hypotenuse, v_x is the adjacent side, and v_y the opposite side. so you can calculate |V| using:

v_x²+V_y²=|V|²

(pythagoras theorem)

So the magnitude is |V|, and is the value of the velocity (e.g. 30m/s), and the direction, θ, is the angle with respect to the x-axis. (e.g 45 degrees).

vectors are relative to a common 'basis' (or axis), and are relative to each other. The components can also be added/subtracted to form a resultant vector, such as is useful for an object with multiple forces on it, amongst other operations too.

Originally Posted by Raymond K

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

If you have a cadd program and you program in Cadd. You know that each point has two inputs. An x and y coordinate.

So if you have a line created by two points. Those points just have to be turned into the hypotenuse of a right triangle.

So if you are given graph coordinates of point one x,y 2,3 and point two x,y 6,7

You would get the square root of ((6-2)^2 plus (7-3)^2) and that would be the length of the hypotenuse of the right triangle formed.

You just need to boil down the x coordinates to the relative distance apart from one another. And do the same for the y coordinates. In this case the triangle formed is 4x4 with a hypotenuse of 5.656854294......

There are four divisions difference in the two x coordinates given and four divisions difference in the two y coordinates given. Basically the square root of A^2 plus B^2

You are just removing the graph and doing a relative measurement of the distance between the two points. Rather then dealing with their origin.

You can then with all sides known and one angle known calculate the angle of the hypotenuse. Or vector if that is what they call it. It would be relative to the graph coordinates.

Sincerely,


William McCormick

Originally Posted by Raymond K

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

I think, this would be better placed in the math section. But anyway ...

Vectors are very useful to evaluate the effect of forces in 3D space. In your example, this would add a velocity component to the car in a direction perpendicular to the driving direction. It is irrelevant for the forward movement, i.e. its component in that direction is zero.

All this is based on the law that any direction can be separated into a number of elementary vectors. The number depends on the dimensions, you want to describe. In 3D you need three vectors (or coordinates) to uniquely define a point in space, in 2D you need two (x/y or latitude/longitude or left/up or radius/angle). In most examples, these base vectors have the length 1 and are perpendicular to each other (but this is not really necessary although it simplifies a lot). This is also the reason for the application of the right triangle, because this visualises the use of orthogonal base vectors. So, in fact, when using vectors, you can evaluate their total impact on other quantities isolated for each dimension individually. Your example demonstrates this nicely.

Thanks, I will refer to this if I ever again have trouble with vectors.

Originally Posted by Dishmaster

Vectors are very useful to evaluate the effect of forces in 3D space. In your example, this would add a velocity component to the car in a direction perpendicular to the driving direction. It is irrelevant for the forward movement, i.e. its component in that direction is zero.

Well, in the example, taking forward motion to be positive x-axis, the wind blowing against the cars motion would be a force vector such as F=<-30,0>, the wind blowing with it would be a force vector, such as F=<30,0>, and the wind blowing sideways on the car would be a force vector like F=<0,30> or F=<0,-30>. Adding the x components together, and the y components together, gives you a resultant vector:

F_r=<30,0>+<-30,0>+<0,30>+<0,-30>
=<30-30+0+0, 0+0+30-30> =<0,0>

In this case the zero vector, meaning resultant force is zero. Just wanted to make clear that the components of a force vector are forces, the components of a velocity vector are velocities etc..

Originally Posted by Dishmaster

Originally Posted by Raymond K

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

I think, this would be better placed in the math section. But anyway ...

Vectors are very useful to evaluate the effect of forces in 3D space. In your example, this would add a velocity component to the car in a direction perpendicular to the driving direction. It is irrelevant for the forward movement, i.e. its component in that direction is zero.

All this is based on the law that any direction can be separated into a number of elementary vectors. The number depends on the dimensions, you want to describe. In 3D you need three vectors (or coordinates) to uniquely define a point in space, in 2D you need two (x/y or latitude/longitude or left/up or radius/angle). In most examples, these base vectors have the length 1 and are perpendicular to each other (but this is not really necessary although it simplifies a lot). This is also the reason for the application of the right triangle, because this visualises the use of orthogonal base vectors. So, in fact, when using vectors, you can evaluate their total impact on other quantities isolated for each dimension individually. Your example demonstrates this nicely.

You need two points to define a single vector (line) in a three dimensional space.

You need x,y,z coordinates for each point. So you will need six inputs.


Sincerely,


William McCormick

Originally Posted by William McCormick

You need two points to define a single vector (line) in a three dimensional space.

You need x,y,z coordinates for each point. So you will need six inputs.

Yeah, right. But since every vector space has an origin, every vector can be related to this. So, a vector has as many components as the magnitude of the space it is located in. A vector connects two points, three components each (in 3D space). But every single point is already constructed by a vector originating from the origin of the coordinate frame.

Originally Posted by Dishmaster

Originally Posted by William McCormick

You need two points to define a single vector (line) in a three dimensional space.

You need x,y,z coordinates for each point. So you will need six inputs.

Yeah, right. But since every vector space has an origin, every vector can be related to this. So, a vector has as many components as the magnitude of the space it is located in. A vector connects two points, three components each (in 3D space). But every single point is already constructed by a vector originating from the origin of the coordinate frame.

That is what x,y,z means.

Coordinates given unless with a specifier, like relative, with a new base point, or relative to last point. Are absolute coordinates, from x,y,z, 0,0,0

Where ever this universal point is or imagined. Usually it is a point on the edge of the plans you are working from. Ha-ha.


Sincerely,


William McCormick

Originally Posted by Raymond K

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

Vectors only try to denote direction and magnitude of physical quantities which possess both of them. They are just arrows drawn on a cartesian reference frame. The arrows denote the direction of the vectors and the magnitude is either denoted by their relative lengths or is specified.Beyond this, I think dealing with them is only upon your mathematical abilities.

Originally Posted by Dishmaster

Originally Posted by William McCormick

You need two points to define a single vector (line) in a three dimensional space.

You need x,y,z coordinates for each point. So you will need six inputs.

Yeah, right. But since every vector space has an origin, every vector can be related to this. So, a vector has as many components as the magnitude of the space it is located in. A vector connects two points, three components each (in 3D space). But every single point is already constructed by a vector originating from the origin of the coordinate frame.

To calculate a three dimensional ray. You need to create the first two dimensional right triangle like we discussed. And then create another triangle using the hypotenuse of the first as one leg of the second right triangle. And the elevation or difference in the two Z coordinates, as the second leg of the triangle. Then you can calculate the actual length of the three dimensional ray.

Sincerely,


William McCormick

Originally Posted by PritishKamat

Originally Posted by Raymond K

We just stated Vectors in Geometry, I can figure them out and work with them, but I don't quite understand what they actually are.

I know an objects velocity is the speed and direction it is going. How does does the word vector relate to velocity, if at all?

If the wind is blowing in the same direction as a car, is that a positive vector? If the wind is blowing against the car, would that be a negative vector? What kind of vector would it be if the wind was blowing to the left or right sides of the car?

Why can you determine the change in direction/magnitude from a right triangle?

Vectors only try to denote direction and magnitude of physical quantities which possess both of them. They are just arrows drawn on a cartesian reference frame. The arrows denote the direction of the vectors and the magnitude is either denoted by their relative lengths or is specified.Beyond this, I think dealing with them is only upon your mathematical abilities.

Today we tend to show elevation on 2D drawings in plan view, with an arrow symbolizing direction and the word "up" or "down" along with an elevation amount. The plan view naturally gives you your other angles and or offsets.

Sincerely,


William McCormick