### Vector calculus in plane and space: Bases, Coordinates and Equations

### Lines in the coordinate space

We have seen that a line #\ell# in the coordinate space #\mathbb{R}^3# can be determined by a parametric representation

\[\rv{x, y, z} = \rv{a, b, c} + \lambda \cdot\rv{u, v, w}

\] wherein #\rv{a, b, c}# is a particular point of the parametric representation, and #\rv{u, v, w}# is a direction vector (at least one of #u#, #v#, #w# does not equal #0#). For each actual value of the parameter #\lambda#, the so-described point #\rv{x,y,z}# belongs to #\ell#, and for each point of #\ell# there is a value of #\lambda#.

A straight line in space can also be described by two linear equations with three unknowns. A line can, in fact, always be seen as the intersection of two planes, which we can imagine through a equation.

From parametric representation to equations for lines and back into the coordinate space

The points of the line #\ell# with parametric representation\[

\rv{x, y, z} = \rv{a, b, c} + \lambda \cdot\rv{u, v, w}

\] are all solutions of the equations you obtain from \[\eqs{x &=& a+\lambda \cdot u\cr y &=& b+\lambda \cdot v\cr z &=& c+\lambda \cdot w\cr }\] by eliminating #\lambda#. By eliminating #\lambda#, each pair gives an equation without #\lambda#. The collection of solutions of these three linear equations with unknown #x#, #y#, #z# is a surface or the whole of space. Two of these three sets of different planes with intersection #\ell#.

Conversely, the solutions of a pair of linear equations with unknown #x#, #y#, #z# form a straight line or a plane. The latter only occurs if the two equations each describe the same plane.

We can achieve the elimination of #\lambda# in the first two equations by subtracting #v# times the first equation from #u# times the second. This gives\[u\cdot y - v\cdot x=u\cdot b-v\cdot a\]In the same manner, we find\[\eqs{u\cdot z-w\cdot x &=& u\cdot c-w\cdot a\cr v\cdot z-w\cdot y &=& v\cdot c-w\cdot b\cr}\]Because at least one of #u#, #v#, #w# does not equal #0#, at least two of the three equations without #\lambda# contain a term with an unknown (these equations cannot be reduced to #0=0#). This means that at least two out of three are planes. In fact, two out of three describe different planes. Because the coordinates of the points on #\ell# satisfy these equations, #\ell# lies in both planes. As two different planes in the space do not intersect at all (in the case of parallel) or intersect in a line, the solutions of the latter two equations form the line #\ell#.

The second statement can be proven with the same geometric observations about the intersection of planes.

Instead of new variable names for each coordinate, we will in future describe the coordinates of a vector #\vec{v}# in #\mathbb{R}^3# as #\rv{v_1,v_2,v_3}#. We will often use the column format. Thus, a parametric representation of a line with particular point #\vec{a}# and direction vector #\vec{v}# in #\mathbb{R}^3# may also be formulated as follows: \[\left(\begin{array}{l}x_1 \\x_2\\x_3

\end{array}\right)

=

\left(\begin{array}{l}

a_1 \\a_2\\a_3

\end{array}\right)

+

\lambda

\left(\begin{array}{l}

v_1 \\v_2\\v_3

\end{array}\right)

\]

Below are examples of the transitions between the parametric representation and equations for a line. Systematic methods to solve these types of calculations, are discussed in the section entitled *Systems of linear equations and matrices*.

Give your answer in the form #a\cdot x+b\cdot y+c\cdot z+d = 0\land f\cdot x+g\cdot y+h\cdot z+j = 0#, where #a#, #b#, #c#, #d#, #f#, #g#, #h#, #j# are integers.

The parametric representation of #\ell# can be written as the system of equations

\[\eqs{x &=& -5 + 4\lambda \cr y &=& 0 -4\lambda \cr z &=& -3 -5\lambda \cr }\]Eliminating #\lambda# from each pair of the three equations gives:

\[\eqs{-4\cdot x-4\cdot y &=&-4\cdot -5-4\cdot 0=20\cr -5\cdot x -4\cdot z&=& -5\cdot -5-4\cdot -3=37\cr -5\cdot y +4\cdot z&=& -5\cdot 0+4\cdot -3=-12\cr}

\]The first two equations determine different planes. They are satisfactory as a system of equations for the line #\ell#. A correct answer is \[-4\cdot x-4\cdot y -20=0\, \land \, -5\cdot x -4\cdot z- 37=0

\]

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