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As noted in the comments, the geometric interpretation of the dot product works well in $\mathbb{R}^2$. I want to sum here this result. We have two vectors $\mathbf{v}= v_1\mathbf{e_1}+v_2\mathbf{e_2}$ and $\mathbf{u}= u_1\mathbf{e_1}+u_2\mathbf{e_2}$ and, being $\{\mathbf{e_1},\mathbf{e_2}\}$ the canonical basis we know that $$ v_1=\mathbf{v}\cos \alpha \quad v_2=\mathbf{v}\sin \alpha $$ and $$ u_1=\mathbf{u}\cos \beta \quad u_2=\mathbf{u}\sin \beta $$ where $\alpha, \beta$ are the angles between $\mathbf{e_1}$ and the two vectors. Now we define $\mathbf{u}\cdot\mathbf{v}=\mathbf{u}\mathbf{v}\cos\theta$ where $\theta=\beta-\alpha$ is the angle between two vectors. Using trigonometry we find $$ \mathbf{u}\mathbf{v}\cos\beta-\alpha=\mathbf{u}\mathbf{v}\cos \alpha \cos \beta+\sin\alpha \sin \beta= $$ $$ =\mathbf{v}\cos \alpha \mathbf{u}\cos\beta+\mathbf{v}\sin \alpha\mathbf{u}\sin \beta= $$ $$ =u_1v_1+u_2v_2 $$ This is fine and give us the machinery to find angles between vectors an to project a vector over an other vector using the components. Generalize this result to any Hilbert space is not so simple, and we have to use the Cauchy-Schwarz inequality that states $$ \left\sum_{i=1}^n{x_iy_i} \right^2\le\sum_{i=1}^n{x_i^2}\sum_{i=1}^n{y_i^2} $$ So, given $\mathbf{x}=\sum_{i=1}{x_i\mathbf{e_i}}$ and $\mathbf{y}=\sum_{i=1}{y_i\mathbf{e_i}}$ we can define the dot product as $$ \mathbf{x}\cdot \mathbf{y}=\sum_{i=1}{x_i y_i} $$ and we have $ \mathbf{x}\cdot \mathbf{y}\le \mathbf{x}\; \mathbf{y} $ So we see that there exists an angle $\theta$ such that $$ \cos \theta = \dfrac{\mathbf{x}\cdot \mathbf{y}}{ \mathbf{x}\; \mathbf{y}} $$ and we can define such $\theta$ as the angle between the two vectors. And , finally, from this we have an analogy with the $\mathbb{R}^2$ situation ad we can find the prjection of a vector on another one. And this work also for infinite dimensional Hilbet spaces. Ces tours, comme d'autres en pisé, conservent encore les vestiges de bandes peintes en blanc sur les zones où [...] de nombreuses couches de pisé ont été [...] superposées, ce qui, vu de loin, pouvait donner l'illusion [...]d'une muraille en pierres de grande taille. This and the other towers made of tabiyya still have traces of the whitewashed bands in the area where the various layers of tabiyya [...] were placed on top of one [...] another, which viewed from a distance make the wall [...]look as though it was built of large blocks of stone. Le groupe d'édifices est [...] magnifique, peu importe de quelle direction on y arrive, et peut être vu de loin. The group of buildings form a most [...] picturesque object from every approach to the city, and can be seen at a great distance. Vu de loin, c'est moins la hauteur [...]vertigineuse que le reflet surnaturel de l'immeuble qui étonne. From a distance, it's not so much [...]its dizzying height as the supernatural reflection of the building that astonishes. Lors de la session Picpus de [...] l'an dernier j'ai vu de loin le convoi d'un [...]amiral, enterré juste à côté de la tombe de La Fayette. During the Picpus program last year [...] from a distance I saw the cortege of an [...]admiral, who was buried just to one side of Lafayette's tomb. Il témoigne nous avons vu de loin un poste douanier de l'armée [...]pakistanaise. He reported "we saw from afar a border post of the Pakistani army. RFI Musique Dix ans après Mar Azul, votre [...] nouvel album São Vicente di longe São Vicente - île dont [...] vous êtes native- vu de loin fait toujours [...]référence au Cap-Vert. RFI Musique Cesaria, your new [...] album Sao Vicente di longe comes a full ten years after Mar Azul, but [...] it's still packed full of references to Cape Verde. Oui, il est [...] certainement superbe vu de loin, mais malheureusement, [...]il bloque presque entièrement notre vision arrière dans le rétroviseur. Oh yes, it might look simply [...] drop dead gorgeous from afar, but unfortunately its [...]positioning almost completely blocks your [...]vision in the rearview mirror. Vu de loin, le pays est animé, radieux [...]et bien organisé. From a distance, it is bustling, [...]bright and together. Tel un joyau étincelant doté d'une crête de verdure, l'édifice est un délicat [...] amalgame de verre, d'acier et de bois surmonté d'un ample toit aux formes [...] naturelles qui, vu de loin, ressemble à [...]un parc. A gleaming jewel with a fuzzy top, the building is an affable mingling of glass, [...] steel and wood under an expansive natural landform roof that makes the building [...] appear more like a park from a distance. Le cuivre joue un double rôle dans cette [...] construction il souligne la forme [...] inhabituelle du bâtiment, même vu de loin, et il le distingue des autres [...]constructions alentour. Copper takes on two special tasks the material [...] highlights to the unusual form of the [...] building even from a distance and sets it apart from the other [...]buildings; and copper's [...]lively surface blends into its natural surroundings. Le logo de [...] l'entreprise s'étale alors en gros caractères sur le toit afin d'être vu de loin. The company [...] logo can be installed in large visible letters on the roof of the building. Rufus a un travail qui [...] peut paraître amusant vu de loin c?est un vendeur de [...]fast food. Rufus has a job the may appear fun from the outside [...]he works at a fast food restaurant. Vu de loin et de tous les côtés, vos produits exercent encore mieux [...]leur pouvoir de séduction. Your products become visible from all sides and afar; the power of seduction [...]becomes dramatic. Allez voir la mine d'or du Superpit, profonde de plus de 400 m et longue de [...] 3,3 km, sous la lumière des projecteurs, avec d'énormes camions qui [...] ressemblent, vus de loin, ? à des jouets [...]pour enfants. See the floodlit superpit goldmine, more [...] than 400 metres deep and over kilometres long, with mammoth machinery that [...] looks like Tonka trucks in the distance. Quand il a vu de loin qu'elle était prise, [...]il a été tout fier de son adresse et il est venu en courant. When he saw from afar that it was trapped, [...]he felt very proud of his skill and ran towards it. Point d'attache pour lampe de sécurité multidirectionnelle [...] SIGNAL pour être vu de loin durant les séances [...]d'entraînement. Attachment point for SiGNAL multidirectional emergency headlamp, allowing [...] user to visible from a distance while training. Vu de loin, le capot de son viseur replié, le Beacon [...]Reflex présente le même aspect général qu'un Brownie Starlet ou un Starflex. When you see it from far, the door of its viewfinder [...]folded, the Beacon Reflex looks like a Brownie Brownie Starlet or a Starflex. Les carreaux de ce [...] tissus sont tellements petits qu'ils donnent l'impression d'un tissu uni vu de loin. The checkered pattern is so thin that you may think the fabric is single-colored. On la voit de loin surgir du sol dans une immense [...]fosse, avec son toit sculpté de croix imbriquées les unes dans les autres. One can see it from a distance jutting out of the [...]earth inside a huge pit with its roof sculpted with interconnected crosses. Design simple et visuel, qui vu de loin donne l'impression d'un personnage [...]qui sourit . Simple design and visual, which seen from afar gives the impression of a character who smiles. Seulement vieux messieurs prennent Opium » Il pensait que j'étais sous les [...] effets de la drogue parce que je lui avais montré mon bras, [...] dont les lésions, vu de loin, ressemblaient à [...]des injections. He thought I was on drugs because I showed [...] him my arm and from that distance it looked [...]like I had injected myself. Il marchait devant ses [...] camarades pour cueillir une fleur qu'il avait vue de loin et soudain a été projeté en l'air, un [...]pied arraché. He went ahead of his friends to a flower which he had seen in the distance and suddenly [...]flew into the air, his foot severed. Ils doivent pouvoir être vus de loin et être perçus également [...]en cas de masquage partiel du colis, notamment lorsqu'il y a de la fumée. They must be able to be seen from a distance, as well as in [...]conditions that are smoky or otherwise partially obscure the package. Un moyen d'être vu de loin, le stop-trottoir est l'outil [...]idéal pour renforcer vos ventes de glaces à l'italienne. One way to be seen from a distance, stop-the sidewalk [...]is the ideal tool for enhancing sales of ice cream in Italian. Vue de loin, après plus de soixante ans, Jubilee apparaît [...]comme une opération fantasque qui n'avait aucune chance de succès, [...]qui ne pouvait qu'entraîner un grand nombre de morts d'hommes. Sixty years later, it seems obvious that Jubilee was a bizarre [...]operation with no chance of success whatsoever and likely [...]to result in a huge number of casualties. Son feuillage extrêmement étroit, [...] panaché de bandes verticales blanches et bleu-vert nous donne, vu de loin, un effet argenté. Fine blades with distinct [...] variegation of green and white stripes give a silvery cast when viewed from a distance. Vu de loin, les 16 mises en scène de lumière [...]réalisées par Hans Peter Kuhn au Patrimoine Culturel Mondial Völklinger Hütte, [...]forment un contraste sans précédent avec les lumières de Saarstahl, l'une des aciéries les plus modernes d'Europe. Observed from a distance Hans Peter Kuhn's [...]16 light scenes at the World Cultural Heritage Site at the Volklingen Ironworks [...]create a unique contrast to the work lights of the Saarstahl plant, one of Europe's most modern steelworks. Vu de loin, avec ses pentes douces et sa forme arrondie, il semble avoir été créé tout [...]spécialement pour faire de Naples [...]la ville la plus harmonieuse du monde. Seen from afar, with its gentle slopes and placid form, Vesuvius seems to bave been [...]created on purpose to make Naples the [...]most harmonious city inn the world, but seen from above, there is na doubt of its terribleness. Je me trouvai, par hasard, [...] devant l'église dont j'avais vu de loin, la veille, les deux tours. By chance, I found [...] myself in front of a church I had seen from afar the previous day, [...]with its two towers. Vu de loin, le volcan semble être couvert par la neige. Seen from the distance, the volcano seems to be covered with snow.

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