Help Linear Algebra,sketching in R^3 and R^4 planes.NEW to forums,urgent help requird

[syedmaheen]

Hey good day everyone,this is my first post here,so I am sorry if I am not following the proper guidelines for posting here.

The following sets of points represent simple geometric figures in a plane.V and B are real numbers.For each problem draw a sketch in the (V,B) plane and a second sketch in the R^2,R^3,R^4 as appropriate.For each problem identify the geometric shape.

a) S={x:x= (0,1) +V(1,2) + B(2,3) for 0<=V<=1,0<=B<=1}
b) S={x:x= V(2,1,-2) + B(4,-2,3) for 0<=V<=6, 0<=B<=V}
c) S={x:x= V(2,1,-2,2) + B(4,-2,3,-1) for 0<=V<=6, 0<=V<=B}

This is first year first semester algebra in my algebra course,and honestly I have no idea how to go about sketching the two diagrams,if someone could break down as to how these should be down in detail,it would be of great help.

[ILikeSerena]

Hey good day everyone,this is my first post here,so I am sorry if I am not following the proper guidelines for posting here.

The following sets of points represent simple geometric figures in a plane.V and B are real numbers.For each problem draw a sketch in the (V,B) plane and a second sketch in the R^2,R^3,R^4 as appropriate.For each problem identify the geometric shape.

a) S={x:x= (0,1) +V(1,2) + B(2,3) for 0<=V<=1,0<=B<=1}
b) S={x:x= V(2,1,-2) + B(4,-2,3) for 0<=V<=6, 0<=B<=V}
c) S={x:x= V(2,1,-2,2) + B(4,-2,3,-1) for 0<=V<=6, 0<=V<=B}

This is first year first semester algebra in my algebra course,and honestly I have no idea how to go about sketching the two diagrams,if someone could break down as to how these should be down in detail,it would be of great help.

Welcome to MHF, syedmaheen!

Did you try to sketch the geometric figure for (a)?
That should be doable, since it's 2-dimensional.

For (b) and (c) you should also sketch 2-dimensional figures.
The only catch is that you need to consider the relation the 2 vectors have to each other in terms of angle and length.

[syedmaheen]

Welcome to MHF, syedmaheen!

Did you try to sketch the geometric figure for (a)?
That should be doable, since it's 2-dimensional.

For (b) and (c) you should also sketch 2-dimensional figures.
The only catch is that you need to consider the relation the 2 vectors have to each other in terms of angle and length.

Yes I did the first one.I did the two vectors on the x and y axis and drew the 4 vertices by plugging in the different values of V and B.
Got a parallelogram,and got the vertices right.
But I was not sure if i did it right.As the problem I am having is,the question states draw in the (v,B) plane and in the R planes,so my question is,which sketch am I doing and how do I do the other sketch?

And honestly,I could not do the sketch in r^3.If you could give a little more detail as to how to solve it.Would be really helpful.

the only catch is that you need to consider the relation the 2 vectors have to each other in terms of angle and length.--how do i find the relation in terms of angle and length and how do i use them?

[ILikeSerena]

Yes I did the first one.I did the two vectors on the x and y axis and drew the 4 vertices by plugging in the different values of V and B.
Got a parallelogram,and got the vertices right.

Yep. That's it.

But I was not sure if i did it right.As the problem I am having is,the question states draw in the (v,B) plane and in the R planes,so my question is,which sketch am I doing and how do I do the other sketch?

And honestly,I could not do the sketch in r^3.If you could give a little more detail as to how to solve it.Would be really helpful.

You can't really draw in 3 or 4 dimensions.
A flat piece of paper just doesn't allow for it.
So you need to sketch a projection.
Since the question asks for the geometric figure, you need to sketch the figure in the plane in which both vectors lie.

the only catch is that you need to consider the relation the 2 vectors have to each other in terms of angle and length.--how do i find the relation in terms of angle and length and how do i use them?

Are you aware of the norm (or length) of a vector? And how to calculate it?
And the dot product?
Or perhaps what it means if 2 vectors point in a direction that is on the same line?

[syedmaheen]

You can't really draw in 3 or 4 dimensions.
A flat piece of paper just doesn't allow for it.
So you need to sketch a projection.
Since the question asks for the geometric figure, you need to sketch the figure in the plane in which both vectors lie.

So basically I use the first two co-ordinates to draw the ones in R^3 and R^4?
And why does the question talk about two sketches?How do I do the OTHER sketch?

Are you aware of the norm (or length) of a vector? And how to calculate it?
And the dot product?
Or perhaps what it means if 2 vectors point in a direction that is on the same line?

Yes I think I am aware of them,length is the square root of the sum of all the squares of all the co-ordinates,am i right?
and dot product between two vectors is the sum of the products of their x,y,z coordinates?i.e., a.b=x1x2 + y1y2 + z1z2 --am i right?

how do i use these factors?
thanks a lot for your replies.and sorry I'm asking so many questions.

[ILikeSerena]

So basically I use the first two co-ordinates to draw the ones in R^3 and R^4?
And why does the question talk about two sketches?How do I do the OTHER sketch?

Now that you mention it, I'm not really sure.
There are actually various ways to draw those sketches.
Perhaps you have more information in your book or notes?

I can hazard a guess though.

I suspect that with a sketch in te (V,B) plane, you are supposed to draw the figure as if both vectors are aligned with the x-axis and the y-axis.
Basically that shows the relationship between the ranges of V and B.
For (a) that means you'd get a 1x1 square.

For a sketch in R2,R3,R4 you can make 3 types of sketches.

First, and what I suspect is intended, a projection in the plane in which the vectors lie.
Perhaps aligning the first vector with the x-axis, and the other with the proper angle and length.

As an alternative, you might sketch the projection on the (X,Y) plane, the (X,Z) plane, and/or the (Y,Z) plane.
You can do that by simply selecting just the X and Y coordinates to draw in the (X,Y) plane.

Finally, as a 3rd option, you could make a drawing in perspective.
But it seems unlikely to me that that is intended.

Yes I think I am aware of them,length is the square root of the sum of all the squares of all the co-ordinates,am i right?
and dot product between two vectors is the sum of the products of their x,y,z coordinates?i.e., a.b=x1x2 + y1y2 + z1z2 --am i right?

how do i use these factors?

Yes.
Additionally the dot product is also the product of the lengths of both vectors times the cosine of the angle between them.
In formula form: .
From this you can deduce the angle between them.