MATH 20C Lecture 3

12.2 3D Space: Surfaces, Vectors, and Curves

Vector Operations

3D Vector Addition

3D Scalar Multiplication

Two nonzero vectors $\vec{v}$ and $\vec{w}$ are parallel iff $\vec{v}=\lambda \vec{w}$ for some scalar $\lambda$

Standard bases in ${\mathbb{R}}^3$

Any 3D vector can be written uniquely as a linearl combination of $\vec{i},\vec{j},\vec{k}$.

Norm/Length/Magnitude of $\vec{v}=⟨a,b,c⟩$

In particular, the distance between is

Vector Parameterization of a Line

is a direction vector of L
Parametric equations of the line L:

Relative Position between two lines in ${\mathbb{R}}^2$

are parallel or coincide for some scalar
intersect at a point
do not intersect and are not parallel (skew lines)