Plane that is constructed by vectors

[mathmari]

Hello!

I found the following in my notes:

The plane that is constructed by two non-parallel vectors $\overrightarrow{v}$ and $\overrightarrow{w}$ consists of all the points of the form $a \overrightarrow{v}+b\overrightarrow{w}$, $a, b \in \mathbb{R}$.

The plane that is defined by $\overrightarrow{v}$ and $\overrightarrow{w}$ is called the plane that is produced by $\overrightarrow{v}$ and $\overrightarrow{w}$.

If $\overrightarrow{v}$ is a multiple of $\overrightarrow{w}$ and $\overrightarrow{w} \neq 0$, then $\overrightarrow{v}$ and $\overrightarrow{w}$ are parallel.

I am asked to find the plane that is produced by the two vectors $\overrightarrow{v_1}=(3, 8, 0)$ and $\overrightarrow{v_2}=(0, 3, 8)$.

Is the plane $a \overrightarrow{v_1}+b\overrightarrow{v_2}$ ?? Or is this only the form of points of the plane?? (Wondering)

 

[I like Serena]

Hello!

I found the following in my notes:

The plane that is constructed by two non-parallel vectors $\overrightarrow{v}$ and $\overrightarrow{w}$ consists of all the points of the form $a \overrightarrow{v}+b\overrightarrow{w}$, $a, b \in \mathbb{R}$.

The plane that is defined by $\overrightarrow{v}$ and $\overrightarrow{w}$ is called the plane that is produced by $\overrightarrow{v}$ and $\overrightarrow{w}$.

If $\overrightarrow{v}$ is a multiple of $\overrightarrow{w}$ and $\overrightarrow{w} \neq 0$, then $\overrightarrow{v}$ and $\overrightarrow{w}$ are parallel.

I am asked to find the plane that is produced by the two vectors $\overrightarrow{v_1}=(3, 8, 0)$ and $\overrightarrow{v_2}=(0, 3, 8)$.

Is the plane $a \overrightarrow{v_1}+b\overrightarrow{v_2}$ ?? Or is this only the form of points of the plane?? (Wondering)

Hey mathmari!

Yep. That is the plane. ;)

I guess that more formally it is indeed the form of a point in a plane.
Since a plane is a set of points, you might make it:
$$\{a \overrightarrow{v_1}+b\overrightarrow{v_2} : a,b \in \mathbb R\}$$
But I consider that nitpicking. (Nerd)

 

[mathmari]

Hey mathmari!

Yep. That is the plane. ;)

I guess that more formally it is indeed the form of a point in a plane.
Since a plane is a set of points, you might make it:
$$\{a \overrightarrow{v_1}+b\overrightarrow{v_2} : a,b \in \mathbb R\}$$
But I consider that nitpicking. (Nerd)

I understand! (Yes)

I have also an other question...

Which is the parallelogram with adjacent sides the vectors $\overrightarrow{w}_1$ and $\overrightarrow{w}_2$ ?? (Wondering)

The set $$\{a \overrightarrow{w_1}+b\overrightarrow{w_2} : 0<a,b <1\}$$

?? (Wondering)

 

[I like Serena]

I understand! (Yes)

I have also an other question...

Which is the parallelogram with adjacent sides the vectors $\overrightarrow{w}_1$ and $\overrightarrow{w}_2$ ?? (Wondering)

The set $$\{a \overrightarrow{w_1}+b\overrightarrow{w_2} : 0<a,b <1\}$$

?? (Wondering)

Yep! (Happy)