### Systems of linear equations and matrices: Systems of Linear Equations

### Planes in space

Just as a line is described in the plane by an equation in two unknowns, a plane in three-dimensional space is described by an equation in three unknowns:

Equation of a plane in three-dimensional space

Let \(a\), \(b\), \(c\), and \(d\) be real numbers. The solution of the equation \[a\cdot x+b\cdot y+c\cdot z+d=0\] with unknown #x#, #y#, and #z#, can be seen as a collection of points in three-dimensional space: it consists of all points \(\rv{x,y,z}\) that satisfy the above equation. If at least one of #a#, #b#, #c# is distinct from #0#, then the solution is a **plane.**

- If \(c\ne0\), then we can write the equation as \(z=-\frac{a}{c}x-\frac{b}{c}y-\frac{d}{c}\). After all, this is the solution if we regard \(x\) and \(y\) as parameters and \(z\) as unknown. It indicates that for each value of \(x\) and \(y\), there is a point \(\rv{x,y,z}\) with \(z\) equal to \(-\frac{a}{c}x-\frac{b}{c}y-\frac{d}{c}\).
- If \(a \ne 0\) or \(b \ne 0\), then we have an
**inclined plane**. - If \(a=0\) and \(b=0\), then the value of \(z\) is always equal to \(-\frac{d}{c}\), and we have a
**horizontal plane**.

- If \(a \ne 0\) or \(b \ne 0\), then we have an
- In the exceptional case \(c=0\), the equation looks like \(a\cdot x+b\cdot y+d=0\).
- If \(a \ne 0\) or \(b \ne 0\), then we have a
**vertical plane**. - If \(a=0\) and \(b=0\) and

- \(d\ne 0\), then there are no solutions;
- \(d=0\), then any triplet of values of \(x\), \(y\), and \(z\) is a solution.

- If \(a \ne 0\) or \(b \ne 0\), then we have a

By describing planes in terms of equations, we are able to find points of intersection of planes as solutions of systems of linear equations.

Intersection of planes The **intersection** of planes (that is, the set of points they have in common intersection) which are described by linear equations in the coordinates #x#, #y#, #z#, can be found by solving the system consisting all of the equations corresponding to these planes.

The intersection of two planes that are not parallel, is a line and can be described with the aid of a single parameter.

If several different planes have a common pont of intersection, then their intersection is a point or a line.

Also, an equation of the plane passing through three different points can be found by solving a system of linear equations:

Plane defined by three points Suppose that three distinct points in \(\mathbb{R}^3\) are represented by the vectors \(\vec{p}=\rv{p_1,p_2,p_3}\), \(\vec{q}=\rv{q_1,q_2,q_3}\) and \(\vec{r}=\rv{r_1,r_2,r_3}\). Then, each plane passing through these three points \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) has a linear equation of the form

\[a\cdot x+b\cdot y+c\cdot z+d=0\] where #\rv{a, b, c,d}# is a solution distinct from #\rv{0,0,0,0}# of the system of linear equations

\[\eqs{ p_1\cdot a+ p_2\cdot b+ p_3\cdot c+d &=&0\\ q_1\cdot a+ q_2\cdot b+ q_3\cdot c+d &=&0\\ r_1\cdot a+ r_2\cdot b+ r_3\cdot c+d &=&0\\ }\]

This can be seen by first substituting #z=0# (the value determined by the third equation) in the other two equations, and next substituting #x=0# (the value determined by the new second equation), and concluding that the first equation now forces #y=0#.

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