Some Handy Expressions in Cartesian Coordinates

(You can watch the video version of this post on YouTube.)

Now there is a well-developed theory of affine transformations with Cartesian coordinates – homogeneous coordinates, actually – by multiplying vectors and matrices (that’s fine), but here are some special cases that are handy to have at the ready.

First of all: if you rotate the vector $(x,y)$ by 90 degrees counterclockwise, then you get $(-y, x)$. If you multiply out the appropriate rotation matrix, this is also what you get, of course, but it keeps things simple to know that it’s really just a swizzle and a minus sign. And it’s obviously linear. Rotation by a variable angle is not a linear transformation, but rotation by any fixed angle is. It’s just that 90 degrees is, like, extra simple.

Let’s talk about the dot product, or inner, or scalar product. It’s just the sum of component-wise products. Simple enough. But it has some nice geometry too. $A\cdot B$ equals the $\lVert A\rVert\cdot\lVert B\rVert\cdot\cos\alpha$, where $\alpha$ is the angle between the vectors. So assuming the vectors are nonzero, their dot product is zero if and only if $A$ and $B$ are perpendicular. It’s positive if they point in the same direction and negative if they point is opposite directions.

We can even get some more geometry by normalising one of the vectors – say B. Then we are left with $\lVert A\rVert \cdot \cos\alpha$: the length of A projected onto B! This can be pretty useful and again, for fixed $B$, it is clearly linear.

Finally, very briefly, there’s the cross product, or outer product. I’ll ignore other dimensions because it gets a bit funky, but in 3D it is super useful. Given two vectors, it gives you a third one that’s perpendicular to both of them and its magnitude has a useful interpretation too: look it up, and while you’re at it, check out why there’s a “left-handed” and a “right-handed” version.