Jul 9, 2025
diff --git a/src/geometry/basic-geometry.md b/src/geometry/basic-geometry.md
 ```

 To see the next important property we should take a look at the set of points $\mathbf r$ for which $\mathbf r\cdot \mathbf a = C$ for some fixed constant $C$.
 You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{|\mathbf a|}$ and they form a hyperplane orthogonal to $\mathbf a$.
 You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{|\mathbf a| ^ 2}$ and they form a hyperplane orthogonal to $\mathbf a$.
 You can see the vector $\mathbf a$ alongside with several such vectors having same dot product with it in 2D on the picture below:

 <div style="text-align: center;">
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Expand Up		@@ -180,7 +180,7 @@ double angle(point2d a, point2d b) {
		```

		To see the next important property we should take a look at the set of points $\mathbf r$ for which $\mathbf r\cdot \mathbf a = C$ for some fixed constant $C$.
		You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{\|\mathbf a\|}$ and they form a hyperplane orthogonal to $\mathbf a$.
		You can see that this set of points is exactly the set of points for which the projection onto $\mathbf a$ is the point $C \cdot \dfrac{\mathbf a}{\|\mathbf a\| ^ 2}$ and they form a hyperplane orthogonal to $\mathbf a$.
		You can see the vector $\mathbf a$ alongside with several such vectors having same dot product with it in 2D on the picture below:

		<div style="text-align: center;">
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