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:

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

Flux

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

$\Phi =\mathrm {comp} _{\vec {N}}{\vec {v}}$

Find the normal of the sphere at point $\left(2,\,{\sqrt {3}},\,0\right)$: ${\vec {N}}=\left\langle 1,\,{\sqrt {3}},\,0\right\rangle$

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

Cross Product

(See MATH 152 Chapter 11.3→)

Given two vectors:

1. ${\vec {v}}=\left\langle v_{x},\,v_{y},\,v_{z}\right\rangle =\left\langle 1,\,2,\,3\right\rangle$
2. ${\vec {w}}=\left\langle w_{x},\,w_{y},\,w_{z}\right\rangle =\left\langle 9,\,8,\,7\right\rangle$

${\begin{aligned}{\vec {v}}\times {\vec {w}}&={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\v_{x}&v_{y}&v_{z}\\w_{x}&w_{y}&w_{z}\\\end{vmatrix}}={\begin{vmatrix}{\hat {\imath }}&{\hat {\jmath }}&{\hat {k}}\\1&2&3\\9&8&7\\\end{vmatrix}}\\&={\hat {\imath }}\left(v_{y}\cdot w_{z}-v_{z}\cdot w_{y}\right)-{\hat {\jmath }}\left(v_{x}\cdot w_{z}-v_{z}\cdot w_{x}\right)+{\hat {k}}\left(v_{x}\cdot w_{y}-v_{y}\cdot w_{x}\right)\\&={\hat {\imath }}\left(2\cdot 7-3\cdot 8\right)-{\hat {\jmath }}\left(1\cdot 7-3\cdot 9\right)+{\hat {k}}\left(1\cdot 8-2\cdot 9\right)\\&=-10{\hat {\imath }}+20{\hat {\jmath }}-10{\hat {k}}\end{aligned}}$

Application: Area of Parallelogram

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

Application: Area of Parallelogram

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

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

${\begin{vmatrix}u_{x}&u_{y}&u_{z}\\v_{x}&v_{y}&v_{z}\\w_{x}&w_{y}&w_{z}\\\end{vmatrix}}$