DLMF:1.6.E9 (Q186): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec

Revision as of 15:57, 1 January 2020

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DLMF:1.6.E9
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    Statements

    Defining formula
    𝐚 × 𝐛 = | 𝐢 𝐣 𝐤 a 1 a 2 a 3 b 1 b 2 b 3 | = ( a 2 ⁢ b 3 - a 3 ⁢ b 2 ) ⁢ 𝐢 + ( a 3 ⁢ b 1 - a 1 ⁢ b 3 ) ⁢ 𝐣 + ( a 1 ⁢ b 2 - a 2 ⁢ b 1 ) ⁢ 𝐤 = ∥ 𝐚 ∥ ⁢ ∥ 𝐛 ∥ ⁢ ( sin ⁡ θ ) ⁢ 𝐧 , 𝐚 𝐛 𝐢 𝐣 𝐤 subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 subscript 𝑎 2 subscript 𝑏 3 subscript 𝑎 3 subscript 𝑏 2 𝐢 subscript 𝑎 3 subscript 𝑏 1 subscript 𝑎 1 subscript 𝑏 3 𝐣 subscript 𝑎 1 subscript 𝑏 2 subscript 𝑎 2 subscript 𝑏 1 𝐤 norm 𝐚 norm 𝐛 𝜃 𝐧 {\displaystyle{\displaystyle\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{% i}&\mathbf{j}&\mathbf{k}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\\ =(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{% 2}-a_{2}b_{1})\mathbf{k}\\ =\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n},}}
    0 references
    DLMF id
    DLMF:1.6.E9
    0 references
    Symbols used
    Q10765
    Defining formula
    det {\displaystyle{\displaystyle\det}}
    xml-id
    C1.S3.SS1.m1adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
     
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