How do you determine if the two lines are parallel, perpendicular, or neither if line a passes through points \[\left( { - 1,4} \right)\] and \[\left( {2,6} \right)\] and line b passes through points \[\left( {2, - 3} \right)\] and \[\left( {8,1} \right)\] ?

Hint: In this question we need to find the relation between the lines formed by the two pairs of points. The two lines in the coordinate plane. If the slope of both the lines are equal, then the two lines are parallel to each other. If the product of the slope of both the lines is \[ - 1\] then, the two lines are perpendicular to each other.
The equation of the line between the two points is obtained as \[y - {y_1} = \left( {\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}} \right)\left( {x - {x_1}} \right)\].

Complete Step By Step solution:
We have given the pair of points are \[\left( { - 1,4} \right)\] and\[\left( {2,6} \right)\]. The other points are \[\left( {2, - 3} \right)\] and\[\left( {8,1} \right)\].
If \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] are the two pair of points, then the slope of these two points is of the form.
\[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Then, the slope of the line between the points \[\left( { - 1,4} \right)\] and\[\left( {2,6} \right)\]is calculated as,
\[
  {m_1} = \dfrac{{6 - 4}}{{2 - \left( { - 1} \right)}} \\
   = \dfrac{2}{3} \\
 \]
Thus, the slope of the first line is \[{m_1} = \dfrac{2}{3}\] .
Then, the slope of the line between the points \[\left( {2, - 3} \right)\] and\[\left( {8,1} \right)\]is calculated as,
\[
  {m_2} = \dfrac{{1 - \left( { - 3} \right)}}{{8 - 2}} \\
   = \dfrac{4}{6} \\
   = \dfrac{2}{3} \\
 \]

The case when the slope of both the lines are equal \[{m_1} = {m_2} = \dfrac{2}{3}\] then the lines are parallel.

Note:
The slope of the line is defined as the rate of change of y coordinates of the lines with respect to the x coordinate of the line. If the slope is positive, then the slope is steeper and is in upward direction. The negative slope is the line that is running downward. The vertical lines have no slope.
The slope intercept form of the line is \[y = mx + c\], here m is the slope and c are the intercept of the line.