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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. dans l'espace far La gare n'est pas très loin d'ici. The station isn't very far from here. loin de a long way from, far from C'est loin d'ici. It's a long way from here. plus loin further plus loin que further than C'est plus loin que la gare. It's further than the station. moins loin not as far moins loin que not as far as dans le temps, futur a long way off , passé a long time ago Noël n'est plus tellement loin. Christmas isn't far off now. Les vacances paraissent déjà tellement loin! The holidays already seem such a long time ago! autres locutions de loin [apercevoir] from a distance, from a long way away On voit l'église de loin. You can see the church from a long way away. Il vient de loin, il doit être fatigué. He's come a long way, he must be tired out. Il vient de loin, ça doit le dépayser. He comes from a long way away, this must be quite a change of scene. fig =de beaucoup by far C'est de loin l'élève la plus brillante. She is by far the brightest pupil. de loin en loin here and there =de temps en temps now and then, every now and then loin de là far from it pas loin de ... presque not far off ... Ça fait pas loin de 1 000 euros. It's not far off 1000 euros. nm au loin in the distance On aperçoit la mer au loin. You can see the sea in the distance. Traduction Dictionnaire Collins Français - Anglais Pour ajouter des entrées à votre liste de vocabulaire, vous devez rejoindre la communauté Reverso. C’est simple et rapide "vu de loin" exemples et traductions en contexte Son personnage royal est habituellement vu de loin. His royal persona is usually seen from afar. Leur mère avait vu de loin les lumières du tracteur s'éteindre. Their mother had seen from afar the lights of the tractor go out. Le nouveau panneau d'affichage du restaurant vu de loin. The new restaurant billboard seen from a distance Si votre affiche est vu de loin, If your poster is seen from a distance, L'étranger pouvait surgir à l'improviste, sans que vous l'ayez vu de loin ni entendu approcher... The stranger could appear unexpectedly, without being seen from far away or heard while walking... Assurez-vous que leur ordinateur est placé de telle sorte que l'écran puisse être vu de loin. Make sure that their computer is placed in such a way that the screen can be seen from far away.

vu to lon dep