Straight Line in Plane

1. General equation of a straight line in the Cartesian coordinate system:
   \(Ax + By + C = 0\),
   where \(x\), \(y\) are the coordinates of a point on the line, \(A\), \(B\), \(C\) are real numbers provided that \({A^2} + {B^2} \ne 0\).
2. Normal vector to a straight line
   Let the line be defined by the general equation
   \(Ax + By + C = 0\),
   Then the vector \(\mathbf{n}\left( {A,B} \right)\) whose coordinates are equal to the coefficients \(A,\) \(B\) is the normal vector to the straight line.

1. Explicit equation of a straight line (slope-intercept form)
   \(y = kx + b.\)
   Here the coefficient \(k = \tan\alpha\) is called the slope of the straight line, and the number \(b\) is the coordinate of intersection of the line with the \(y\)-axis.

1. The slope of a straight line is determined by the formula
   \(k = \tan \alpha =\) \( {\large\frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}\normalsize},\)
   where \(A\left( {{x_1},{y_1}} \right)\), \(B\left( {{x_2},{y_2}} \right)\) are the coordinates of two points of the line.

1. Equation of a straight line given a point and a slope (point-slope form)
   \(y = {y_0} + k\left( {x – {x_0}} \right)\),
   where \(k\) is the slope, and the point \(P\left( {{x_0},{y_0}} \right)\) lies on the straight line.

1. Equation of a straight line passing through two points (two-point form)
   \({\large\frac{{y – {y_1}}}{{{y_2} – {y_1}}}\normalsize} = {\large\frac{{x – {x_1}}}{{{x_2} – {x_1}}}\normalsize}\)  or  \(\left| {\begin{array}{*{20}{c}}
   x & y & 1\\
   {{x_1}} & {{y_1}} & 1\\
   {{x_2}} & {{y_2}} & 1
   \end{array}} \right| = 0.\)

1. Intercept form of a straight line equation
   \({\large\frac{x}{a}\normalsize} + {\large\frac{y}{b}\normalsize} = 1\),
   where \(a\) and \(b\) are defined as \(x\)-intercept and \(y\)-intercept, respectively.

1. Normal form of a straight line equation
   \(x\cos \beta + y\sin \beta \) \(-\; p \) \(= 0\)
   Here \(\cos \beta\) and \(\sin\beta =\) \( \cos \left( {90^\circ – \beta} \right)\) are the direction cosines of the normal vector. The parameter \(p\) is equal to the distance between the straight line and the origin.

1. Point direction form of a straight line equation
   \({\large\frac{{x – {x_1}}}{X}\normalsize} = {\large\frac{{y – {y_1}}}{Y}\normalsize}\),
   where the vector \(\mathbf{s}\left( {X,Y} \right)\) is directed along the straight line, and the point \(P\left( {{x_1},{y_1}} \right)\) lies on the line. This equation is also called the standard or canonical equation of the straight line.

1. Vertical line equation
   \(x= a\)
2. Horizontal line equation
   \(y= b\)
3. Vector equation of a straight line
   \(\mathbf{r}= \mathbf{a} + t\mathbf{b}\),
   where the vector \(\mathbf{a}\) is drawn from the origin to a known point \(A\) lying on this line. The vector \(\mathbf{b}\) determines the direction of the straight line. The vector \(\mathbf{r} = \mathbf{OX}\) is the position vector directed from the origin to any point \(X\) on this line. The number \(t\) is a parameter that varies from \( – \infty \) to \(\infty \).

1. Equation of a straight line in parametric form
   \(
   \left\{
   \begin{aligned}
   x &= {a_1} + t{b_1} \\
   y &= {a_2} + t{b_2}
   \end{aligned}
   \right.
   \),
   where \(\left( {{a_1},{a_2}} \right)\) are the coordinates of a known point \(A\) lying on this line, (\(\left( {x,y} \right)\) are the coordinates of an arbitrary point of the line, \(\left( {{b_1},{b_2}} \right)\) are the coordinates of a vector \(\mathbf{b}\) parallel to the given straight line, \(t\) is a parameter.

1. Distance from a point to a straight line
   The distance \(d\) from the point \(M\left( {{x_1},{y_1}} \right)\) to the line \(Ax + By + C \) \(= 0\) is given by the formula
   \(d = {\large\frac{{\left| {A{x_1} + B{y_1} + C} \right|}}{{\sqrt {{A^2} + {B^2}} }}\normalsize}.\)

1. Parallel lines
   Two straight lines \(y = {k_1}x + {b_1}\) and \(y = {k_2}x + {b_2}\) are parallel if
   \({k_1} = {k_2}\).
   Two straight lines \({A_1}x + {B_1}y + {C_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2} \) \(= 0\) are parallel if
   \({\large\frac{{{A_1}}}{{{A_2}}}\normalsize} = {\large\frac{{{B_1}}}{{{B_2}}}\normalsize}\).

1. Perpendicular lines
   Two straight lines \(y = {k_1}x + {b_1}\) and \(y = {k_2}x + {b_2}\) are perpendicular if
   \({k_1} = – {\large\frac{1}{{{k_2}}}\normalsize}\) or equivalently \({k_1}{k_2} = – 1\).
   Two straight lines \({A_1}x + {B_1}y + {C_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2} \) \(= 0\) are perpendicular if
   \({A_1}{A_2} + {B_1}{B_2} = 0\).

1. Angle between straight lines
   \(\tan \varphi = {\large\frac{{{k_2} – {k_1}}}{{1 + {k_1}{k_2}}}\normalsize},\;\) \(\cos \varphi = {\large\frac{{{A_1}{A_2} + {B_1}{B_2}}}{{\sqrt {A_1^2 + B_1^2} \sqrt {A_2^2 + B_2^2} }}\normalsize}\)

1. Intersection of two lines
   If the two straight lines \({A_1}x + {B_1}y + {C_1} \) \(= 0\) and \({A_2}x + {B_2}y + {C_2} \) \(= 0\) intersect, then the coordinates of the intersection point are
   \({x_0} = {\large\frac{{ – {C_1}{B_2} + {C_2}{B_1}}}{{{A_1}{B_2} – {A_2}{B_1}}}\normalsize},\;\) \({y_0} = {\large\frac{{ – {A_1}{C_2} + {A_2}{C_1}}}{{{A_1}{B_2} – {A_2}{B_1}}}\normalsize}.\)