How to Visualize Vectors and Subspaces in a Coordinate System?

In summary: Wondering)In summary, the conversation discusses how to visualize a vector in a coordinate system using a given basis. It is suggested to draw the three vectors of the basis and the vector itself, as well as the multiples of the basis vectors used in the linear combination. The conversation also discusses how to sketch a subspace in a coordinate system by finding the intersections with each coordinate plane. The use of a program like Geogebra or TikZ is recommended to create these visualizations.
  • #1
mathmari
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Hey! :eek:

We have the basis $B=\left \{\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix},\begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}1 \\ 2 \\ 1\end{pmatrix} \right \}$ of $\mathbb{R}^3$ and the vector $v$ can we written as a linear combination of the elements of the basis as follows:
\begin{equation*}v=\begin{pmatrix}2 \\ 8 \\ -1\end{pmatrix}=-\frac{17}{2}\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}+\frac{3}{2}\begin{pmatrix}2 \\ 1\\ 0\end{pmatrix}+\frac{15}{2}\begin{pmatrix}1 \\ 2\\ 1\end{pmatrix}\end{equation*}

It is asked to visualize that in a coordinate system, by showing how the vector $v$ is generated by the basis.

For that do we draw the three vectors of the basis and the vector $v$ or do we have to draw also the multiples of the elements of the basis that we use at the linear combination? (Wondering)

I mean, is the following enough?

View attachment 8601

(Wondering)
Further, we have the subspace $U=\left \{\begin{pmatrix}a \\ a+b \\ 2a-b\end{pmatrix}\middle| a, b\in \mathbb{R}\right \}$ of $\mathbb{R}^3$. How could we sketch $U$ in a coordinate system? (Wondering)
 

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  • #2
mathmari said:
It is asked to visualize that in a coordinate system, by showing how the vector $v$ is generated by the basis.

For that do we draw the three vectors of the basis and the vector $v$ or do we have to draw also the multiples of the elements of the basis that we use at the linear combination? (Wondering)

I mean, is the following enough?

That's a bit subjective.
Since it says 'showing how the vector $v$ is generated by the basis', I'm inclined to say that we should do a bit more.
So yes, we could draw also the multiples of the elements of the basis that we use at the linear combination.
Or alternatively, in addition to the basis itself, we could draw all the lines of the parallellepipedum.

Here's a couple of examples how we might do it:
300px-3D_Vector.svg.png

img149.png


mathmari said:
Further, we have the subspace $U=\left \{\begin{pmatrix}a \\ a+b \\ 2a-b\end{pmatrix}\middle| a, b\in \mathbb{R}\right \}$ of $\mathbb{R}^3$. How could we sketch $U$ in a coordinate system? (Wondering)

Here's a couple of examples how to sketch a plane.

Draw a rectangle with a couple of vectors to show how the vectors relate to the plain:
image001.png


Draw the intersections with the coordinate planes:
View attachment 8602
 

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  • plane_by_coordinate_planes.png
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  • #3
Klaas van Aarsen said:
That's a bit subjective.
Since it says 'showing how the vector $v$ is generated by the basis', I'm inclined to say that we should do a bit more.
So yes, we could draw also the multiples of the elements of the basis that we use at the linear combination.
Or alternatively, in addition to the basis itself, we could draw all the lines of the parallellepipedum.

Which way do you suggest? The one with the parallellepipedum?

And how did you make these sketches? Is there a programm for that?

(Wondering)
Klaas van Aarsen said:
Here's a couple of examples how to sketch a plane.

Draw a rectangle with a couple of vectors to show how the vectors relate to the plain:Draw the intersections with the coordinate planes:

Could you explain to me further these two sketches? (Wondering)
 
  • #4
mathmari said:
Which way do you suggest? The one with the parallellepipedum?

And how did you make these sketches? Is there a program for that?

I just copied them after googling.

And sure, there are plenty of programs to make sketches like these.
Didn't you already use one for post #1?
We can probably find a free online one.

Nowadays I would do it with Latex myself (TikZ).
Like this:
\begin{tikzpicture}[>=stealth']
%preamble \usepackage{tikz-3dplot}
%preamble \usetikzlibrary{arrows}
\tdplotsetmaincoords{60}{100}
\begin{scope}[scale=0.6,tdplot_main_coords]
\coordinate (O) at (0,0,0);
\coordinate (P) at (2,8,-1);
\coordinate (A) at ({-17/2},{-17/2},{-17/2});
\coordinate (B) at ({3},{3/2},{0});
\coordinate (C) at ({15/2},{15},{15/2});

\draw[help lines,->] (O) -- (8,0,0) node[anchor=north east]{$x$};
\draw[help lines,->] (O) -- (0,8,0) node[anchor=north west]{$y$};
\draw[help lines,->] (O) -- (0,0,8) node[anchor=south]{$z$};

\draw[->] (O) -- (A);
\draw[->] (A) -- ++(B);
\draw[->] (A) ++(B) -- ++(C);

\draw[ultra thick,->] (O) -- (1,1,1) node[anchor=south west]{$b_1$};
\draw[ultra thick,->] (O) -- (2,1,0) node[anchor=north west]{$b_2$};
\draw[ultra thick,->] (O) -- (1,2,1) node[anchor=south]{$b_3$};

\draw[ultra thick,->,color=red] (O) -- (P) node[anchor=west] {$v$};
\end{scope}
\begin{scope}[scale=0.6,xshift=15cm,tdplot_main_coords]
\coordinate (O) at (0,0,0);
\coordinate (P) at (2,8,-1);
\coordinate (A) at ({-17/2},{-17/2},{-17/2});
\coordinate (B) at ({3},{3/2},{0});
\coordinate (C) at ({15/2},{15},{15/2});

\draw[->] (O) -- (8,0,0) node[anchor=north east]{$x$};
\draw[->] (O) -- (0,8,0) node[anchor=north west]{$y$};
\draw[->] (O) -- (0,0,8) node[anchor=south]{$z$};

\draw[help lines] (O) -- (A) -- +(B) (A) -- ++(C) -- (P);
\draw[help lines] (O) -- (B) -- +(C) (B) -- ++(A) -- (P);
\draw[help lines] (O) -- (C) -- +(A) (C) -- ++(B) -- (P);

\draw[ultra thick,->] (O) -- (1,1,1) node[anchor=south west]{$b_1$};
\draw[ultra thick,->] (O) -- (2,1,0) node[anchor=north west]{$b_2$};
\draw[ultra thick,->] (O) -- (1,2,1) node[anchor=south]{$b_3$};

\draw[ultra thick,->,color=red] (O) -- (P) node[anchor=west] {$v$};
\end{scope}
\end{tikzpicture}

mathmari said:
Could you explain to me further these two sketches? (Wondering)

For the first we would draw the 2 vectors that are in the plane.
And then draw a rectangle around it aligned with those vectors.
Since your plane intersects the origin, there is no need to draw a vector from the origin to the plane. (Thinking)

For the second, we would need to figure out the intersections of the plane with each of the coordinate planes.
That is, solve for $x=0$, respectively $y=0$, respectively $z=0$.
That should yield 3 vectors, which we can draw as lines.
We can color the area where it's between the coordinate axes as shown in the picture. (Thinking)
 
  • #5
Klaas van Aarsen said:
And sure, there are plenty of programs to make sketches like these.
Didn't you already use one for post #1?
We can probably find a free online one.

Nowadays I would do it with Latex myself (TikZ).

Ah ok! In the first post I used Geogebra, so if I use this program I will draw the three elements of the basis $b_1, b_2, b_3$, then $-\frac{17}{2}b_1$, $\frac{3}{2}b_2$, $\frac{15}{2}b_3$ and then the sum of these ones? (Wondering)
Klaas van Aarsen said:
For the first we would draw the 2 vectors that are in the plane.
And then draw a rectangle around it aligned with those vectors.
Since your plane intersects the origin, there is no need to draw a vector from the origin to the plane. (Thinking)
So, we draw the two vectors of the basis $\begin{pmatrix}1 \\ 1\\ 2\end{pmatrix}, \begin{pmatrix}0 \\ 1\\ -1\end{pmatrix}$, then we draw a rectangle that passes through the two endpoints of these vectors and the origin, since it is contained in $U$.

Have I understood that correctly? (Wondering)
 
  • #6
mathmari said:
Ah ok! In the first post I used Geogebra, so if I use this program I will draw the three elements of the basis $b_1, b_2, b_3$, then $-\frac{17}{2}b_1$, $\frac{3}{2}b_2$, $\frac{15}{2}b_3$ and then the sum of these ones?

I see you can use [M]Vector(<start point>, <end point>)[/M].
It means that we can also start the second vector at the end of the first one, can't we?

And if we want to draw a parellellepipedum, we can use [M]Line(<point>, <point>)[/M], can't we? (Wondering)

Or else we can indeed draw the vectors you mentioned. (Doh)

mathmari said:
So, we draw the two vectors of the basis $\begin{pmatrix}1 \\ 1\\ 2\end{pmatrix}, \begin{pmatrix}0 \\ 1\\ -1\end{pmatrix}$, then we draw a rectangle that passes through the two endpoints of these vectors and the origin, since it is contained in $U$.

Have I understood that correctly?

I'd draw a rectangle, actually a rhombus, that is a bit bigger, but yeah, that's basically it.

Btw, did you try [M]Plane(...)[/M] in geogebra?
It pops up 6 different methods to define a plane... and then geogebra will draw it for you! (Muscle)
 
  • #7
Klaas van Aarsen said:
I see you can use [M]Vector(<start point>, <end point>)[/M].
It means that we can also start the second vector at the end of the first one, can't we?

And if we want to draw a parellellepipedum, we can use [M]Line(<point>, <point>)[/M], can't we? (Wondering)

Or else we can indeed draw the vectors you mentioned. (Doh)

So, you mean to draw the vector $-\frac{17}{2}b_1$, then at the endpoint of that one we start the vector $\frac{3}{2}b_2$, then at the endpoint of that one we start the vector $\frac{15}{2}b_3$ and then we draw a vector from the origin to the endpoint of the last vector.

Or did you mean something else? (Wondering)
Klaas van Aarsen said:
I'd draw a rectangle, actually a rhombus, that is a bit bigger, but yeah, that's basically it.

Btw, did you try [M]Plane(...)[/M] in geogebra?
It pops up 6 different methods to define a plane... and then geogebra will draw it for you! (Muscle)

Ok! (Nerd)
 
  • #8
mathmari said:
So, you mean to draw the vector $-\frac{17}{2}b_1$, then at the endpoint of that one we start the vector $\frac{3}{2}b_2$, then at the endpoint of that one we start the vector $\frac{15}{2}b_3$ and then we draw a vector from the origin to the endpoint of the last vector.

Or did you mean something else?

(Nod)
 

Related to How to Visualize Vectors and Subspaces in a Coordinate System?

What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. It is represented by an arrow pointing in the direction of the vector, with its length representing the magnitude.

How are vectors represented in a coordinate system?

Vectors are typically represented in a coordinate system using two numbers: the x-coordinate and the y-coordinate. These numbers indicate the direction and magnitude of the vector in relation to the axes of the coordinate system.

What is the difference between a position vector and a displacement vector?

A position vector represents the location of a point in relation to the origin of a coordinate system. A displacement vector represents the change in position from one point to another.

How are vectors added and subtracted?

To add vectors, you can place them tip-to-tail, and the resulting vector will be the one connecting the tail of the first vector to the tip of the last vector. To subtract vectors, you can place them head-to-head, and the resulting vector will be the one connecting the head of the first vector to the head of the second vector.

What is the difference between a scalar and a vector?

A scalar is a mathematical quantity that has only magnitude, while a vector has both magnitude and direction. Examples of scalars include temperature and mass, while examples of vectors include velocity and force.

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