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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3 = \left\{ \left( a,\, b,\, c \right) ~|~ a,\, b,\, c, \in \mathbb{R} \right\}}

Flux

Suppose the sphere Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x-1 \right)^2 + y^2 + z^2 = 4} is in a fluid flowing at velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \hat\imath - 2 \hat\jmath} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi = \mathrm{comp}_{\vec{N}} \vec{v}}

Find the normal of the sphere at point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( 2,\, \sqrt{3},\, 0 \right)} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{N} = \left\langle 1,\, \sqrt{3},\, 0 \right\rangle}

Therefore, the flux is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \left\langle v_x,\, v_y,\, v_z \right\rangle = \left\langle 1,\, 2,\, 3 \right\rangle}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w} = \left\langle w_x,\, w_y,\, w_z \right\rangle = \left\langle 9,\, 8,\, 7 \right\rangle }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}}

Application: Area of Parallelogram

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \vec{v} \times \vec{w} \right|} is the area of the parallelogram spanned by the vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}}

Application: Area of Parallelogram

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u} \times \vec{v} \cdot \vec{w}} represents the volume of the parallelepiped

Even simpler, the determinent of the matrix formed by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{vmatrix} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \\ \end{vmatrix}}