Lecture 003

Surfaces in $\mathbb{R}^3$

Spheres: $(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$ Planes: $ax + by + cz = d$ (planes are a special type of cylinder, a line stretch out) Cylinders: $x^2 + y^2 = 1$, or $z = 4x^2$, or something involve $x, y, z$ (a curve in 2D that stretch infinitely or finitely in one linear direction)

Dot Product

Dot Product: given $\overrightarrow{u} = \langle{u_1, ..., u_n}\rangle$ and $\overrightarrow{v} = \langle{v_1, ..., v_n}\rangle$, the dot (inner, scalar) product is the scalar $\overrightarrow{u} \cdot \overrightarrow{v} = u_1v_1 + ... + u_nv_n$

Dot Product operate on same: $\overrightarrow{u} \cdot \overrightarrow{u} = \|\overrightarrow{u}\|^2$

Physicist's Dot Product: (dot product gives the angle) for $\overrightarrow{u}$ and $\overrightarrow{v}$ with $0 \leq \theta \leq \pi$ as the angle, then $\overrightarrow{u} \cdot \overrightarrow{v} = \|\overrightarrow{u}\| \|\overrightarrow{v}\| \cos{\theta}$

Property of Dot Product

1. $u \cdot v = v \cdot u$
2. $u \cdot (v + w) = u \cdot v + u\cdot w$
3. $C(u \cdot v) = (Cu) \cdot v = u \cdot (Cv)$
4. $v \cdot v = \|v\|^2$

Law of Cosines: $C^2 = A^2 + B^2 - 2AB\cos{c}$

Orthogonal: iff $\overrightarrow{u} \cdot \overrightarrow{v} = 0$ Projection of Vectors: projection of $\overrightarrow{v}$ onto $\overrightarrow{u}$ is $proj_{\overrightarrow{u}}(v) = (\cos{\theta} \cdot \|v\|) \cdot \frac{\overrightarrow{u}}{\|u\|} = \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\|\overrightarrow{u}\|^2}\cdot \overrightarrow{u}$ Scalar Projection: $\|comp_{\overrightarrow{u}}(v)\| = \|proj_{\overrightarrow{u}}(v)\| = \frac{|\overrightarrow{u}\cdot \overrightarrow{v}|}{\|\overrightarrow{u}\|}$ (dot product can be negative, but output is positive, so signore sign)