Using colored dots has the advantage of being able to do projections and sections with the minimum modification of the formula. A better color scheme would...
This is at least a visual demonstration, if not an analytical proof, that of all shapes of rectangular solids, the cube is the one will maximize the volume for...
Work in progress towards using eigenvectors to align 2nd degree conic sections to the axes. Would like to find an approach which generalizes to aligning cubic...
... will do it. I'm not sure why GC demands the "if" clause here, and won't allow the other form of clipping. On Mar 14, 2013, at 5:11 AM, Bo Johannesson <...
Work in progress showing that a sort of generalized Pythagorean relationship comes out of linear algebra on Euclidean space. Just as the projections of a line...
Hej! Would it be possible to use Subscript in Summation and directly get the numerical result as it is in an example with Dot Product? Is it OK to use i as a...
Ron, In the example below with the column vector times the row vector, isn't it inconsistent to require the dot for this, since it's really just a kind of...
I'm still having the same problem with row and column vectors when the components are _not_ elements of a table. At least for the case when the variables are...
Hej! Is my attached DotProduct.gcf a good way to show how to, in Graphing Calculator, apply the algebraic definition and illustrate the example from Wikipedia...
Hej! I have made a document which illustrates two vectors in the x,y-plane and the angle between them. The way I do to plot the arc between the vectors is...
The determinant always equals . This can be seen from the diagram above, where the length of MB times the length of VA gives the area of the parallelogram on...
The determinant formula and the inner product formula (in 2D Euclidean space) together form 2 functional equations which I think uniquely specify the sine and...
Actually, maybe it's more intuitive to get our definitions of the inner and outer products (= determinant, in the 2D case) from the projection functions we use...
... If you just want the small angle between vectors drawn from the origin, it's enough to use . But, even then, we've got to correct for the case when the...
The idea is that we have to prevent each succeeding vector from projecting on any of the previously orthogonalized vectors. Then we're guaranteed to have an...
Non-square matrices are either "embedding" or "detaching" mappings. This one, , takes the vector that follows and "embeds" it in the subspace of xyz-space...
[OK, I got a little carried away by my frustration at the total lack of interest I'm getting from everyone on what I think are exciting discoveries. Not...
Below, I've used non-square matrices in two ways: once, via , to map the gray circle in the 2D view to the slanted plane in the 3D view, and secondly, via the...
[I forgot to include the expressions for the circles and dots in the previous posting.] Below, I've used non-square matrices in two ways: once, via , to map...
If we use the matrix-product versions of projections onto a line and a plane, we save some computation. It's also visually clearer. We Just back off on each...
Here, we streamline the orthogonalization by using the formula for projection onto the plane determined by unit vectors u and v: We use the matrix identity . ...
If we restrict the test point to the z = 0 plane, then we have the same coordinates for its projection in the projected grid, as we'd expect for a linear...
The projection of a cube onto the diagonal plane specified by x + y + z = 0 will yield a hexagon with each of the parallelograms taking up two segments. ...
