Whit.a.6.1 Show that the plane H defined by:

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In summary, the plane H defined by:$H=\left\{\alpha_1\left[\begin{array}{rrr}1\\1\\1\end{array} \right]+\alpha_2\left[\begin{array}{rrr}1\\-1\\0\end{array} \right]\textit{ Such that } \alpha_1,\ \alpha_1\in\mathbb{R}\right\}$=\begin{bmatrix}a_1+a_2\\ a_1+a_2\\ a_1\end{bmatrix}$$\text{
  • #1
karush
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$\tiny{whit.a.6.1}$

Show that the plane H defined by:

$H=\left\{
\alpha_1\left[
\begin{array}{rrr}1\\1\\1\end{array} \right]
+\alpha_2\left[\begin{array}{rrr}1\\-1\\0\end{array} \right]
\textit{ Such that } \alpha_1,\ \alpha_1\in\mathbb{R}\right\}
=\begin{bmatrix}a_1+a_2\\ a_1+a_2\\ a_1\end{bmatrix}$
$\text{rref}(H)=\left[ \begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 0 \end{array} \right]$
ok I don't know what this answers
 
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  • #2
What question are you trying to answer?

The fact that
$H=\left\{
\alpha_1\left[
\begin{array}{rrr}1\\1\\1\end{array} \right]
+\alpha_2\left[\begin{array}{rrr}1\\-1\\0\end{array} \right]
\textit{ Such that } \alpha_1,\ \alpha_1\in\mathbb{R}\right\}$
IS the same as
$\begin{bmatrix}a_1+a_2\\ a_1+a_2\\ a_1\end{bmatrix}$
should be obvious from the definitions of "scalar multiplication" and "addition of vectors".

Or are you trying to say that something is true about that plane? If so, what?

($\left[ \begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 0 \end{array} \right]$ doesn't appear to have anything to do with this problem and "$\text{rref}(H)$" doesn't make sense because "H" is plane, not a matrix.)
 
  • #3
It's odd that Karush "liked" my response but did not answer the questions I asked.
 
  • #4
Country Boy said:
What question are you trying to answer?

The fact that
$H=\left\{
\alpha_1\left[
\begin{array}{rrr}1\\1\\1\end{array} \right]
+\alpha_2\left[\begin{array}{rrr}1\\-1\\0\end{array} \right]
\textit{ Such that } \alpha_1,\ \alpha_1\in\mathbb{R}\right\}$
IS the same as
$\begin{bmatrix}a_1+a_2\\ a_1+a_2\\ a_1\end{bmatrix}$
should be obvious from the definitions of "scalar multiplication" and "addition of vectors".

Or are you trying to say that something is true about that plane? If so, what?

($\left[ \begin{array}{cc|c} 1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & 0 \end{array} \right]$ doesn't appear to have anything to do with this problem and "$\text{rref}(H)$" doesn't make sense because "H" is plane, not a matrix.)
that was shown in an example
 

Related to Whit.a.6.1 Show that the plane H defined by:

1. What is the definition of the plane H?

The plane H is defined by a linear equation in three variables, typically written in the form ax + by + cz + d = 0, where a, b, and c are not all zero. This equation represents all points (x, y, z) that lie on the plane.

2. How is the plane H different from other planes?

The plane H is unique because it is defined by a specific linear equation. Other planes may have different equations and therefore represent different sets of points in space.

3. Can you give an example of a plane H?

One example of a plane H could be the xy-plane, which is defined by the equation z = 0. This plane contains all points with a z-coordinate of 0.

4. How many points are needed to define a plane H?

A plane H can be defined by any three non-collinear points. This means that the points cannot all lie on the same line.

5. How is the plane H used in mathematics and science?

The plane H is used in various mathematical and scientific fields, including geometry, physics, and engineering. It is often used to represent surfaces or boundaries in three-dimensional space and can be used to solve equations and analyze geometric relationships.

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