matematika



Formulas from plane analytic geometry

Distance d between two pointsP_1(x_1 \textrm{ , } y_1) and P_2(x_2 \textrm{ , } y_2)

fig 1
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Slope m of line joining two points P_1(x_1 \textrm{ , } y_1) and P_2(x_2 \textrm{ , } y_2)

m = \frac{y_2 - y_1}{x_2 - y_1} = \textrm { tan } \theta

Equation of line joining two points P_1(x_1 \textrm{ , } y_1) and P_2(x_2 \textrm{ , } y_2)

\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - y_1} = m \qquad \qquad \qquad \qquad \qquad \qquad \textrm{ or } \qquad \qquad \qquad \qquad \qquad \qquad y - y_1 = m(x - x_1) \\ y = mx + b
where b = y_1 - mx_1 = \frac{x_2y_1 - x_1y_2}{x_2 - x_1} is the intercept on the y axis, i.e. y intercept.

Equation of line in terms of x intercept a \ne 0 and y intercept b \ne 0

fig 2
\frac{x}{a} + \frac{y}{b} = 1

Normal form for equation of line

x \textrm{ cos } \alpha + y \textrm{ sin } \alpha = p
where p = perpendicular distance from origin O to line
and     α = angle of inclination of perpendicular with
with positive x axis.
fig 3

General equation of line

Ax + By + C = 0

Distance from point (x_1 \textrm{ , } y_1) to line Ax + By + C = 0

\frac{Ax_1 + By_1 + C}{\pm \sqrt{A^2 + B^2}}
where the sign is chosen so that the distance is nonnegative.

Angle \psi between two lines having slopes m_1 and m_2

\textrm{ tan } \psi = \frac{m_2 - m_1}{1 + m_1m_2}
Lines are parallel or coincident if and only if m_1 = m_2.
Lines are perpendicular of and only if m_2 = -\frac{1}{m_1}.
fig 4

Area of triangle with vertices at (x_1 \textrm{ , } y_1) \textrm{ , } (x_2 \textrm{ , } y_2) \textrm{ , } (x_3 \textrm{ , } y_3)

Area
 = \pm \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{array} \right| \\ = \pm y_2 ( x_1y_2 + y_1x_3 + y_3x_2 - y_2x_3 - y_1x_2 - x_1y_3 )
where the sign is chosen so that the area is nonnegative. If the area is zero the points all lie on a line.
fig 5


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