@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-NMDH2N1P-9
  skos:definition """En mathématiques, et plus précisément en algèbre et en géométrie vectorielle, le <b>produit scalaire</b> est une opération algébrique s'ajoutant aux lois s'appliquant aux vecteurs. C'est une forme bilinéaire, symétrique, définie positive. À deux vecteurs, elle associe un scalaire, c'est-à-dire un nombre tel que ceux qui définissent cet espace vectoriel — réel pour un espace vectoriel réel. Dans un espace vectoriel E sur le corps ℝ des nombres réels, le produit scalaire de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {u}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {u}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce70b04c12b0575de787a71dac4239e911e9cdf6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\overrightarrow {u}}}"></span> par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {v}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {v}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbb1a68c861cbd0cfda4f71510f67eed27c7cb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\overrightarrow {v}}}"></span> est le scalaire (l'élément de ℝ), noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo>⋅<!-- ⋅ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7056dea4ebbbeee27cad7fb25182cfe60012fbe2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.456ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle ({\\\\overrightarrow {u}}|{\\\\overrightarrow {v}})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ({\\\\overrightarrow {u}}|{\\\\overrightarrow {v}})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99eb2a98906763561b1f9295c5b3a8ae4f8ca94" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:3.509ex;" alt="{\\\\displaystyle ({\\\\overrightarrow {u}}|{\\\\overrightarrow {v}})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}}|{\\\\overrightarrow {v}}\\
angle }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}}|{\\\\overrightarrow {v}}\\
angle }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5e814aeb8be007be9ed12555d6c053808076a6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.233ex; height:3.509ex;" alt="{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}}|{\\\\overrightarrow {v}}\\
angle }"></span>, ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}},{\\\\overrightarrow {v}}\\
angle }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo>,</mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}},{\\\\overrightarrow {v}}\\
angle }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2215136114634271811cb1d97a0522335e34412e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.62ex; height:3.509ex;" alt="{\\\\displaystyle \\\\langle {\\\\overrightarrow {u}},{\\\\overrightarrow {v}}\\
angle }"></span>. Le produit scalaire est donné par : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}=||{\\\\overrightarrow {u}}||\\	imes ||{\\\\overrightarrow {v}}||\\	imes \\\\cos({\\\\widehat {{\\\\overrightarrow {u}},{\\\\overrightarrow {v}}}})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo>⋅<!-- ⋅ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mo>×<!-- × --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mo stretchy="false">|</mo>         </mrow>         <mo>×<!-- × --></mo>         <mi>cos</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow class="MJX-TeXAtom-ORD">             <mover>               <mrow>                 <mrow class="MJX-TeXAtom-ORD">                   <mover>                     <mi>u</mi>                     <mo>→<!-- → --></mo>                   </mover>                 </mrow>                 <mo>,</mo>                 <mrow class="MJX-TeXAtom-ORD">                   <mover>                     <mi>v</mi>                     <mo>→<!-- → --></mo>                   </mover>                 </mrow>               </mrow>               <mo>^<!-- ^ --></mo>             </mover>           </mrow>         </mrow>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}=||{\\\\overrightarrow {u}}||\\	imes ||{\\\\overrightarrow {v}}||\\	imes \\\\cos({\\\\widehat {{\\\\overrightarrow {u}},{\\\\overrightarrow {v}}}})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768e6459c010f63905628727766d0493664c8919" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.917ex; height:4.509ex;" alt="{\\\\displaystyle {\\\\overrightarrow {u}}\\\\cdot {\\\\overrightarrow {v}}=||{\\\\overrightarrow {u}}||\\	imes ||{\\\\overrightarrow {v}}||\\	imes \\\\cos({\\\\widehat {{\\\\overrightarrow {u}},{\\\\overrightarrow {v}}}})}"></span>, c'est-à-dire le produit des normes des vecteurs <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {u}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>u</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {u}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce70b04c12b0575de787a71dac4239e911e9cdf6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\overrightarrow {u}}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\\\overrightarrow {v}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mover>             <mi>v</mi>             <mo>→<!-- → --></mo>           </mover>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\overrightarrow {v}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbb1a68c861cbd0cfda4f71510f67eed27c7cb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.388ex; height:3.009ex;" alt="{\\\\displaystyle {\\\\overrightarrow {v}}}"></span> par le cosinus de l'angle formé par les deux vecteurs. Le produit scalaire permet d'exploiter les notions de la géométrie euclidienne traditionnelle : longueurs, angles, orthogonalité en dimension deux et trois, mais aussi de les étendre à des espaces vectoriels réels de toute dimension, et (avec certaines modifications dans la définition) aux espaces vectoriels complexes.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Produit_scalaire">https://fr.wikipedia.org/wiki/Produit_scalaire</a>)"""@fr, """In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Dot_product">https://en.wikipedia.org/wiki/Dot_product</a>)"""@en ;
  skos:broader psr:-R556XDWR-W ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Dot_product>, <https://fr.wikipedia.org/wiki/Produit_scalaire> ;
  a skos:Concept ;
  skos:altLabel "dot product"@en ;
  skos:prefLabel "scalar product"@en, "produit scalaire"@fr ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date .

psr:-R556XDWR-W
  skos:prefLabel "algebraic operation"@en, "opération algébrique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-NMDH2N1P-9 .