GraphingCalcUsers at Yahoo! Groups http://groups.yahoo.com/group/GraphingCalcUsers/ User group for Pacific Tech's Graphing Calculator math education visualization software Hypercube projected onto diagonal hyperplane. Generalized cross-prod Mon, 06 May 2013 10:09:17 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1978 http://groups.yahoo.com/group/GraphingCalcUsers/message/1978 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. Vector function for oblique projections. Restricting test point to z Wed, 01 May 2013 12:47:05 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1977 http://groups.yahoo.com/group/GraphingCalcUsers/message/1977 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 Oblique projection onto plane, vector function Wed, 01 May 2013 12:41:07 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1976 http://groups.yahoo.com/group/GraphingCalcUsers/message/1976 This is based on the derivation at http://math.stackexchange.com/questions/117418/projection-to-the-plane-in-the-direction-of-the-vector Gram-Schmidt orthogonalization. Shortcut via projection onto plane. Fri, 12 Apr 2013 18:39:02 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1975 http://groups.yahoo.com/group/GraphingCalcUsers/message/1975 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 . Orthogonalization of a basis (Gram-Schimidt and adapted) Wed, 10 Apr 2013 17:11:57 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1974 http://groups.yahoo.com/group/GraphingCalcUsers/message/1974 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 Non-square matrices used to both embed and detach a vector mapping, Tue, 09 Apr 2013 05:15:38 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1973 http://groups.yahoo.com/group/GraphingCalcUsers/message/1973 [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 Non-square matrices used to both embed and detach a vector mapping Tue, 09 Apr 2013 05:11:29 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1972 http://groups.yahoo.com/group/GraphingCalcUsers/message/1972 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 Visualizing non-square matrices, cont. Embedding in a sub-space. Mon, 08 Apr 2013 14:04:37 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1971 http://groups.yahoo.com/group/GraphingCalcUsers/message/1971 [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 Visualizing non-square matrices, cont. Mon, 08 Apr 2013 13:59:26 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1970 http://groups.yahoo.com/group/GraphingCalcUsers/message/1970 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 Vector geometry of Gram-Schmidt procedure in 3D Wed, 03 Apr 2013 14:38:54 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1969 http://groups.yahoo.com/group/GraphingCalcUsers/message/1969 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 Re: Arc and quadrants [1 Attachment] Fri, 22 Mar 2013 14:18:16 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1968 http://groups.yahoo.com/group/GraphingCalcUsers/message/1968 ... 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 Functional equations for trig functions Fri, 22 Mar 2013 14:13:30 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1967 http://groups.yahoo.com/group/GraphingCalcUsers/message/1967 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 Determinant and inner product as related to trig angle addition form Fri, 22 Mar 2013 14:07:53 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1966 http://groups.yahoo.com/group/GraphingCalcUsers/message/1966 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 Arc function. Relation of determinant to sign of angle. Determinant Fri, 22 Mar 2013 12:51:03 GMT Christopher Young http://groups.yahoo.com/group/GraphingCalcUsers/message/1965 http://groups.yahoo.com/group/GraphingCalcUsers/message/1965 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 Arc and quadrants Fri, 22 Mar 2013 10:24:11 GMT Bo Johannesson http://groups.yahoo.com/group/GraphingCalcUsers/message/1964 http://groups.yahoo.com/group/GraphingCalcUsers/message/1964 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