DIRECTION COSINES — A
DIRECTION COSINES — B
UNIT VECTOR â
UNIT VECTOR b̂
LINE — VECTOR FORM
CARTESIAN FORM
A+B via parallelogram rule.
✦ 3D GEOMETRY — LINE & PLANE ✦
1 · EQUATION OF A LINE
A line is defined by:
• A point A(x₁, y₁, z₁)
• A direction vector d = ⟨a, b, c⟩
Vector form:
r = a + λd
r = position vector of any point on the line
a = position vector of the given point
d = direction vector · λ = parameter
Cartesian (symmetric) form:
(x−x₁)/a = (y−y₁)/b = (z−z₁)/c
2 · EQUATION OF A PLANE
A plane is defined by:
• A point A(x₁, y₁, z₁)
• A normal vector n = ⟨a, b, c⟩
Vector form:
(r − a) · n = 0
Cartesian form:
a(x−x₁) + b(y−y₁) + c(z−z₁) = 0
or, after expanding:
ax + by + cz + d = 0
where d is a constant.
✦ EASY WAY TO REMEMBER
Line = Point + Direction
r = a + λd
Plane = Point + Normal
(r − a) · n = 0
1-finger: rotate/pan
2-finger: zoom + pan
Drag A or B tip to move vectors
Formula:
A=(3,2)
+
B=(1,3)
=
R=(4,5)