Analytical Geometry

Rectangular coordinate system labels

x–coordinate = abscissa; y–coordinate = ordinate; z–coordinate = applicate

Complex plane (Argand plane)

rectangular form z = a + bi, where a is the real part (x–component) and b is the imaginary part (y–component)

Polar form

z = r∠theta, where r is magnitude/distance and theta is argument/angle

Rectangular to polar coordinates

r = sqrt(x^2 + y^2) and theta = tan^–1(y/x)

Polar to rectangular coordinates

x = r cos(theta) and y = r sin(theta)

Distance between two points in a plane

d = sqrt((x2 – x1)^2 + (y2 – y1)^2)

Distance between two points in space

d = sqrt((x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2)

Distance between a point and a line

d = |Ax1 + By1 + C| / sqrt(A^2 + B^2) for line Ax + By + C = 0

Distance between two parallel lines

d = |C2 – C1| / sqrt(A^2 + B^2) for lines Ax + By + C1 = 0 and Ax + By + C2 = 0

Slope fundamental equations

m = rise/run = Delta y / Delta x = tan(theta)

Slope formula between two points

m = (y2 – y1) / (x2 – x1)

General equation of a line

Ax + By + C = 0

Point slope form

y – y1 = m(x – x1)

Slope intercept form

y = mx + b

Two point form

y – y1 = [(y2 – y1) / (x2 – x1)] * (x – x1)

Intercept form

x/a + y/b = 1

Parallel lines condition

m1 = m2

Perpendicular lines condition

m1 * m2 = –1

Acute angle between two lines

theta = tan^–1 |(m2 – m1) / (1 + m1 * m2)|

Acute angle between two lines (General Form)

theta = tan^–1 |(a2b1 – a1b2) / (a1a2 + b1b2)|

Acute angle between two planes

cos(theta) = |n1 · n2| / (|n1| * |n2|) = (A1A2 + B1B2 + C1C2) / [sqrt(A1^2+B1^2+C1^2) * sqrt(A2^2+B2^2+C2^2)]

Midpoint formula

(xM, yM) = ((x1 + x2) / 2, (y1 + y2) / 2)

Internal section formula

C(x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))

Shoelace Formula (Area by Coordinates)

Area = (1/2) |(x1y2 + x2y3 + … + xny1) – (y1x2 + y2x3 + … + ynx1)|