*When dealing with planes in a 3D space, one must not only know how to solve 3x3 linear systems (most of which dependent on a given variable ** or ** generally) but also how to interpret the results found. Each condition (the system being inconsistent or having infinitely many solutions) has a geometrical meaning. We'll see a couple of examples in the exercise below.*

## Question

**[Maximum marks: 12]**

Consider the system below where each equation represents a plane in a 3D space:

**a.** Reduce the system to row echelon form using row operations **[4 marks]**

**b.** In the following exercises:

**i.** Find the values of for which the system has infinitely many solutions. Interpret this result geometrically. **[2 marks]**

**ii.** Find the values of for which the system has no solutions. Interpret this result geometrically. **[2 marks]**

**c.** Find the unique solution in terms of . **[2 marks]**

**d.** Find the cartesian equation of the line which crosses the intersection between the three planes when perpendicularly to the plane . **[2 marks]**

## Solution

**a.** We'll start reducing the linear system to row echelon form. By doing so, we'll be able to simplify the system and analise the conditions that would make it have one solution, infinite solutions or no solution (all three possibilities will show up in this example). The sooner we start, the sooner we finish.

After simplifying the rows, one get the following:

Multiplying the third row by we get

The system is equivalent to:

**b. i.** The equivalent system is

**For **

The equivalent system is

By setting and substituting it into the second row, we get . By substituting both on the first row, . The solution is a line of intersection between the three planes (none of which is parallel). The parametric equations of the line found are:

**ii.**

**For **

The equivalent system is

We can see from rows 2 and 3 that these planes are parallel. The first one, shown by the first row of the linear system, is not parallel to the other two. Therefore, we have two parallel planes and an intersection one, where the intersection of the first plane and any of the two forms a line parallel to the third.

**c.** We can find a unique solution for all values of different than and .

**For **

Isolating on the second row gives us

Lastly, isolating on the first row gives us

Therefore, the unique solution (as a function of , is a point given by the coordinates

**d.** When the unique solution will be

The line's direction vector is the same as the plane's normal vector. Substituting into the plane's equation,

Click on the animation below lalala

Click on the animation below lalala

Click on the animation below lalala