Formulas and Tables
Analytic Geometry
Straight Line in Plane
Point coordinates: \(x,\) \({x_0},\) \({x_1},\) \({x_2},\) \(y, \ldots \)
Real numbers: \(k,\) \(a,\) \(b,\) \(p,\) \(t,\) \(A,\) \(B,\) \(C,\) \({A_1},\) \({A_2}, \ldots\)
Angle between two lines: \(\varphi\)
Angles: \(\alpha\), \(\beta\)
Direction vectors: \(\mathbf{s}\), \(\mathbf{b}\)
Normal vector: \(\mathbf{n}\)
Position vectors: \(\mathbf{a}\), \(\mathbf{r}\)
Distance from a point to a line: \(d\)
- 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 \({A^2} + {B^2} \ne 0\). - 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.
- 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.
- 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.
- 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.
- 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 \(\left| {\begin{array}{*{20}{c}}
x & y & 1\\
{{x_1}} & {{y_1}} & 1\\
{{x_2}} & {{y_2}} & 1
\end{array}} \right| = 0.\)
- 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.
- 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.
- 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.
- Vertical line equation
\(x= a\) - Horizontal line equation
\(y= b\) - 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 \(\infty \).
- 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.
- 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
\(d = {\large\frac{{\left| {A{x_1} + B{y_1} + C} \right|}}{{\sqrt {{A^2} + {B^2}} }}\normalsize}.\)
- 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
\({\large\frac{{{A_1}}}{{{A_2}}}\normalsize} = {\large\frac{{{B_1}}}{{{B_2}}}\normalsize}\).
- 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
\({A_1}{A_2} + {B_1}{B_2} = 0\).
- Angle between straight lines
\(\tan \varphi = {\large\frac{{{k_2} – {k_1}}}{{1 + {k_1}{k_2}}}\normalsize},\;\) \(\cos \varphi = {\large\frac{{{A_1}{A_2} + {B_1}{B_2}}}{{\sqrt {A_1^2 + B_1^2} \sqrt {A_2^2 + B_2^2} }}\normalsize}\)
- 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
\({x_0} = {\large\frac{{ – {C_1}{B_2} + {C_2}{B_1}}}{{{A_1}{B_2} – {A_2}{B_1}}}\normalsize},\;\) \({y_0} = {\large\frac{{ – {A_1}{C_2} + {A_2}{C_1}}}{{{A_1}{B_2} – {A_2}{B_1}}}\normalsize}.\)