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Vectors

Chapter 2 · Lines & Planes

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Vector equation of a line
r=a+λd\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}
a\mathbf{a} = point on line, d\mathbf{d} = direction, λ∈R\lambda\in\mathbb{R}.
Parallel lines
Direction vectors are scalar multiples.
Point on a line
Substitute and solve for a single consistent λ\lambda.
Equation of a plane
n⋅r=d  ⇔  n1x+n2y+n3z=d\mathbf{n}\cdot\mathbf{r} = d \;\Leftrightarrow\; n_1x + n_2y + n_3z = d
Normal vector
The coefficients of x,y,zx,y,z ARE the normal n\mathbf{n}; it is ⟂ to the plane.
Finding d
d=n⋅ad = \mathbf{n}\cdot\mathbf{a}
Using any known point a\mathbf{a} on the plane.
Line (vector form)
r=a+td\mathbf r = \mathbf a + t\mathbf d
a = point on line, d = direction.
Plane
r⋅n=a⋅n\mathbf r\cdot\mathbf n = \mathbf a\cdot\mathbf n
n = normal vector.
Intersections
Lines: solve for t,s (parallel if directions ∥; skew if no solution in 3D).
Common trap
Two lines can be skew (never meet, not parallel) in 3D — no 2D analogue.