MATH 251 Lecture 3

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3D Vectors (cont'd)

(See MATH 152 Chapter 11.2→)

Define 3D space as all tuples of length 3 with real numbers:

${\mathbb{ℝ}}^3=\{(a,b,c)|a,b,c,\in \mathbb{ℝ}\}$

Flux

Suppose the sphere ${(x-1)}^2+y^2+z^2=4$ is in a fluid flowing at velocity $3\hat{ı}-2\hat{ȷ}$.

$\mathrm{\Phi }={\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}_{\overrightarrow{N}}\overrightarrow{v}$

Find the normal of the sphere at point $(2,\sqrt{3},0)$: $\overrightarrow{N}=⟨1,\sqrt{3},0⟩$

Therefore, the flux is $\mathrm{\Phi }=\frac{\overrightarrow{v}\cdot \overrightarrow{N}}{\left|\overrightarrow{N}\right|}=\frac{3}{2}-\sqrt{3}$

Cross Product

(See MATH 152 Chapter 11.3→)

Given two vectors:

1. $\overrightarrow{v}=⟨v_x,v_y,v_z⟩=⟨1,2,3⟩$
2. $\overrightarrow{w}=⟨w_x,w_y,w_z⟩=⟨9,8,7⟩$

$\begin{array}{rl}\overrightarrow{v}\times \overrightarrow{w}& =\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ v_x& v_y& v_z\\ w_x& w_y& w_z\\ \end{array}\right|=\left|\begin{array}{ccc}\hat{ı}& \hat{ȷ}& \hat{k}\\ 1& 2& 3\\ 9& 8& 7\\ \end{array}\right|\\ & =\hat{ı}(v_y\cdot w_z-v_z\cdot w_y)-\hat{ȷ}(v_x\cdot w_z-v_z\cdot w_x)+\hat{k}(v_x\cdot w_y-v_y\cdot w_x)\\ & =\hat{ı}(2\cdot 7-3\cdot 8)-\hat{ȷ}(1\cdot 7-3\cdot 9)+\hat{k}(1\cdot 8-2\cdot 9)\\ & =-10\hat{ı}+20\hat{ȷ}-10\hat{k}\end{array}$

Application: Area of Parallelogram

$|\overrightarrow{v}\times \overrightarrow{w}|$ is the area of the parallelogram spanned by the vectors $\overrightarrow{v}$ and $\overrightarrow{w}$

Application: Area of Parallelogram

$\overrightarrow{u}\times \overrightarrow{v}\cdot \overrightarrow{w}$ represents the volume of the parallelepiped

Even simpler, the determinent of the matrix formed by $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$

$\left|\begin{array}{ccc}{u}_x& u_y& u_z\\ v_x& v_y& v_z\\ w_x& w_y& w_z\\ \end{array}\right|$