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)

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