Linear Geometry
3.1 Equation of a Straight Line in Three Dimensions
Parametric and Cartesian equations of lines in 3D, direction vectors, distance between skew lines, angle between lines, intersection of lines, distance from point to line.
3.1 Equation of a Straight Line in Three Dimensions
Vector (parametric) form: A line through point A with direction vector d:
r = a + td (t ∈ ℝ)
Components: x = a₁+td₁, y = a₂+td₂, z = a₃+td₃
r = a + td (t ∈ ℝ)
Components: x = a₁+td₁, y = a₂+td₂, z = a₃+td₃
Cartesian (symmetric) form:
x−a₁d₁ = y−a₂d₂ = z−a₃d₃ = t
x−a₁d₁ = y−a₂d₂ = z−a₃d₃ = t
Example: Line through A(1,2,3) with direction d = 4i+5j+6k:
Parametric: x=1+4t, y=2+5t, z=3+6t
Cartesian: x−14 = y−25 = z−36
Parametric: x=1+4t, y=2+5t, z=3+6t
Cartesian: x−14 = y−25 = z−36
Line through two points A and B:
d = b−a (direction from A to B)
r = a + t(b−a) = (1−t)a + tb
d = b−a (direction from A to B)
r = a + t(b−a) = (1−t)a + tb
3.1.3 Angle Between Two Lines
cos θ = |d₁·d₂||d₁||d₂|
Use absolute value — angle between lines is always 0° to 90°.
3.1.4 Intersection of Two Lines
Set r₁ = r₂, solve for parameters s and t. If consistent → intersect. If inconsistent → skew lines.
3.1.1 Distance Between Two Skew Lines
d = |(a₂−a₁)·(d₁×d₂)||d₁×d₂|
Example: L₁: r=(1,0,−1)+t(1,1,1), L₂: r=(0,2,0)+s(2,1,−1)
d₁×d₂ = |i,j,k;1,1,1;2,1,−1| = (−2)i+(3)j+(−1)k
|d₁×d₂| = 4+9+1 = 14
a₂−a₁ = (−1,2,1)
d = |(−1)(−2)+(2)(3)+(1)(−1)|14 = |2+6−1|14 = 714 = 142
d₁×d₂ = |i,j,k;1,1,1;2,1,−1| = (−2)i+(3)j+(−1)k
|d₁×d₂| = 4+9+1 = 14
a₂−a₁ = (−1,2,1)
d = |(−1)(−2)+(2)(3)+(1)(−1)|14 = |2+6−1|14 = 714 = 142
3.1.5 Distance from a Point to a Line
d = |AP×d||d|
where A is a point on the line, P is the external point, d is direction vector
where A is a point on the line, P is the external point, d is direction vector