Cartesian Vectors

[sysD]

Three vectors:

 

Vector A = (2,-5,-1)

 

Vector B = (2,0,-1)

 

Vector C = (-1,0,1)

 

What does it mean to have an entire vector represented by cartesian coordinates? Do those points represent the head of the arrow?

I know that squaring the components and rooting the entirety yields the magnitude of the vector, but can someone explain what these points represent?

BTW Vector A can also be written as (a) with a right-facing arrow on top.

 

 

In context, these three vectors are going to be used to define the volume of a parallelepiped.

[studiot]

Yes you got it.

 

The three numbers are the (x,y,z) coordinates of the head of the arrow for each vector.

 

You can draw each vector as a line from the origin to each point. The direction of the vector is the same as the sense from the origin (0,0,0) to that particular point.

 

Geometrically they can be thought of as three edges of a parallelepiped that meet at a vertex placed at the origin.

 

They are examples of what is known as the position vector. This describes a line from the origin to any point in space by a vector, thus allowing us to define any point in space by a vector and any set of points by a set of position vectors.

 

Being able to describe any point in space in this way allows us to use vector algebra and calculus to calculate useful things like the volume of the parallelepiped.

 

However I suggest you get a good hold of these basics before going on to this.

[sysD]

Thanks, that helped a lot.

 

 

Would the volume of the parallelepiped be calculated by:

 

V= | (a x b)dot w |

 

 

?

 

I tried graphing out those vectors but it doesn't look like the typical basis of a parallelepiped... the angle between A and C is obtuse.

Edited September 8, 2012 by sysD

[studiot]

You are certainly getting the correct formulae from somewhere, but is it not explaining what is going on?

 

Any two vectors that intersect (meet at a point) define a plane.

If you draw a parallelogram in that plane with the two vectors forming two of the sides

 

The cross product of these two vectors gives the area of that parallelogram.

 

The third vector converts the parallelogram to a parallelepiped with volume equal to the perpendicular distance between the base area and the slant height (which is the length of the third vector).

 

This, of course is the dot product of the third vector with the cross product of the other two.

Edited September 8, 2012 by studiot

[sysD]

Okay, I think I understand.

I'm still trying to wrap my head around orienting structures in three dimensions.

Also, we don't cover planes until the next chapter .

 

I have a related problem - mind checking to see if my answer is correct?

 

A parallelepiped is defined by the following vectors:

a=(2,-5,-1)

b=(2,0,-1)

c=(-1,0,1)

Calculate the volume.

 

My answer:

(a x b)dot c

=(5,0,10) dot (-1,0,1)

=(-5+0+10)

=5

 

The volume is 5 units cubed.

 

 

I checked my cross product answer using the dot product check method... I don't know of any methods to check the answer yielded by a dot product... so I'm posting here

[studiot]

Yes I make it that as well.

 

[sysD]

Thanks.