Three Dimensional Geometry - Vocabulary Flashcards

Vocabulary-style flashcards covering key concepts from three-dimensional geometry and lines in space.

Distance from the y-axis

The shortest distance from a point P(a,b,c) to the y-axis equals sqrt(a^2 + c^2).

Distance from the x-axis

The shortest distance from P(a,b,c) to the x-axis equals sqrt(b^2 + c^2).

Distance from the z-axis

The shortest distance from P(a,b,c) to the z-axis equals sqrt(a^2 + b^2).

Direction cosines

The cosines of the angles between a line and the positive x-, y-, z-axes: (cos α, cos β, cos γ); satisfy cos^2 α + cos^2 β + cos^2 γ = 1.

Direction ratios

A triple (a,b,c) giving the direction of a line; the line’s direction vector is proportional to (a,b,c).

Vector equation of a line

r = r0 + λ d, where r0 is a point on the line and d is the direction vector.

Line through two points (Cartesian form)

Line through P1(x1,y1,z1) and P2(x2,y2,z2) has direction d = P2−P1; parametric form r = P1 + t d; symmetric form (x−x1)/dx = (y−y1)/dy = (z−z1)/dz.

Perpendicular lines (3D)

Two lines with direction ratios (a1,b1,c1) and (a2,b2,c2) are perpendicular if a1a2 + b1b2 + c1c2 = 0.

Parallel lines

Two lines are parallel if their direction ratios are proportional: (a1,b1,c1) ∝ (a2,b2,c2).

Shortest distance between skew lines

d = |(a2−a1) · (b1 × b2)| / |b1 × b2|, where a1,a2 are points on the lines and b1,b2 their direction vectors.

Angle between two lines

If direction vectors are b1 and b2, then cos θ = |b1 · b2| / (|b1||b2|).

Line parallel to the z-axis through (a,b,c)

The line x = a and y = b (z is free); it is parallel to the z-axis.

Line through a point parallel to a given line

If a line passes through a point A with direction vector d, its equation is r = A + λ d.

Line through two points (vector form)

Line through P and Q: r = P + t (Q − P).

Foot of perpendicular

The point where the perpendicular from a given point to a line meets the line.

Distance between parallel lines

If L1: r = a1 + t b and L2: r = a2 + t b share direction b, then d = |(a2 − a1) × b| / |b|.

Direction cosines for equal angles with axes

A line making equal angles with x, y, z axes has direction cosines proportional to (1,1,1); normalized to (1/√3,1/√3,1/√3) up to sign.

Line through a point parallel to a given line (rephrase)

Through a and parallel to direction vector d: r = a + λ d.

Intersection of two lines

Two lines intersect if there exist parameters λ and μ such that r1(λ) = r2(μ).

Line through a point perpendicular to two lines

Direction is given by the cross product of the two lines’ direction vectors.

Line through a point parallel to z-axis (explicit form)

x = a, y = b defines a line parallel to the z-axis passing through (a,b, any z).

Vector equation of a line through a and parallel to b

r = a + λ b, where a is a point on the line and b is its direction vector.

Distance from a point to a line (3D)

d = |(P0 − P1) × d| / |d|, where P0 is the point, P1 is a point on the line, and d is the line’s direction vector.

Direction ratios vs direction cosines (relation)

Direction cosines are the normalized components of the direction vector: (l,m,n) with l^2 + m^2 + n^2 = 1; direction ratios (a,b,c) relate by (l,m,n) ∝ (a,b,c)/√(a^2+b^2+c^2).