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There are several ways to attack this problem.The first is to quite simply invoke Euler's rotation theorem, which states that any finite number of rotations around a single fixed point (but around arbitrary axes in [math]n[/math] dimensions) can be expressed as a single rotation of angle [math]\theta [/math]around an axis [math]\hat{n}[/math].If we accept that every rotation is represented by a matrix, and that the method of rotating a vector is matrix multiplication, then it follows immediately from this that the product of rotation matrices [math]A_1 A_2 ... A_n[/math] must also be a rotation matrix — else we have violated Euler’s rotation theorem.The question is, of course, how you actually prove this theorem.Euler’s original work is…gross. It involves many, many triangles drawn on the surface of spheres (i.e. non-Euclidean triangles).If you fancy following the proof through to the end, the wikipedia page linked earlier seems to do a halfway decent job.An alternative method (or, equivalently, a secondary way to prove Euler’s theorem, I guess), is to directly use the properties of rotation matrices, with a little excursion into Group Theory.A rotation, mathematically speaking, is any operation in which the distances between all points in the space remains constant, and which leaves a points, or set of points, fixed (assuming we're on a simple Euclidean space), in addition to preserving the orientation structure of the object.In group-theoretic language, we call these operations (on Euclidean space) the “Special Orthogonal Group in [math]n[/math] dimensions”, or [math]SO(n)[/math] for short.The matrices [math]A[/math][math][/math] which are members of [math]SO(n)[/math] are defined by the following two properties:[math]A^T A = 1_n[/math] (the ‘orthogonal’ bit)[math]\text{det}(A) = 1[/math] (the ‘special’ bit)I.e. rotation matrices are orthogonal matrices with determinant one. Here [math]1_n[/math] is the identity matrix in [math]n[/math] dimensions.The “orthogonality” condition is the condition that ensures that distances are preserved, since in Euclidean space we have the length [math]d[/math]of a vector [math]\mathbf{v}[/math] being:[math]\displaystyle d^2 = \mathbf{v} \cdot \mathbf{v} \tag*{}[/math]If we rotate this vector, such that [math]\mathbf{v}^\prime = [/math][math]A[/math][math] \mathbf{v},[/math] with [math]A[/math][math][/math] an orthogonal vector, then, by the properties of matrix multiplication:[math]\displaystyle \mathbf{v}^\prime \cdot \mathbf{v}^\prime = \mathbf{v}^T A^T [/math][math]A[/math][math] \mathbf{v} = \mathbf{v}^T \mathbf{v} = d^2 \tag*{}[/math]Hence, the distance [math]d[/math] was unaffected by the rotation.All orthogonal matrices have determinant [math]\pm 1,[/math] but those with a negative determinant also include a mirror-flip reflection around some axis. Given that we want pure rotations, not reflections, in order to preserve the orientation of objects in our space, we therefore restrict ourselves to those with positive determinants — which is where the “special” bit comes from.The fact that I have mentioned that these structures form a Group (with the associated operation being matrix multiplication) is in fact sufficient to conclude that the product of several rotation matrices is in fact also a rotation, since groups are defined to be closed under the group operation.This means that any two elements [math]g_1[/math] and [math]g_2[/math] in a group [math]G,[/math] with group operation [math]g_1 \bullet g_2[/math] must return a third element, [math]g_3,[/math] which is also a member of the group [math]G[/math]. Therefore if [math]A[/math][math][/math] and [math]B[/math][math][/math] are rotation matrices, from the definition of a group, it follows that [math]A[/math][math] \bullet [/math][math]B[/math][math] = AB[/math] is also a rotation matrix.Of course…this is a cheat way out. In order to state that [math]SO(n)[/math] was a group, I need to prove that this was true! It can also be shown explicitly from the following general properties of the transpose and determinant:[math](AB)^T = B^T A^T[/math][math]\text{det}(AB) = \text{det}(A)\text{det}(B)[/math]We therefore construct a matrix [math]C = [/math][math]A[/math][math] [/math][math]B[/math][math],[/math] where [math]A[/math][math][/math] and [math]B[/math][math][/math]are members of [math]SO(n).[/math]We then consider:[math]C^T C = (B^T A^T)(A B) = B^T (A^T A) B = B^T B = {1_n}[/math]Since [math]A^T A = 1_n [/math]and [math]B^T B = 1_n[/math], and the associativity of matrix multiplication.[math]\text{det}(C) = \text{det}(AB) = \text{det}(A) \text{det}(B) = 1 \times 1 = 1[/math]We therefore see that [math]C[/math] is an orthogonal matrix, with determinant one — i.e. it is a member of [math]SO(n),[/math] and thus is a rotation matrix.We have therefore proved that [math]SO(n)[/math](and indeed [math]O(n)[/math]) forms a group that is closed under matrix multiplication, and hence, by definition, the concatenation of multiple rotations is in itself a rotation.We have therefore proved Euler’s Rotation Theorem.

It’s a very old joke from the 50s BBC comedy show The Goon Show. A joke that can only work on radio.The Goon Show is still fondly remembered, mostly by English blokes of a certain age, mostly older than me—it went off the air years before I was born, but I discovered it as a child. It was a surrealist explosion, a half hour of madness broadcast once a week in the evenings, lasting from 1951 to 1960. The Beatles were all fans of it, and you can hear its influence on their Christmas records and sometimes on their albums and songs: ‘Yellow Submarine’ and ‘You Know My Name (Look Up The Number)’ are very Goon-like.I discovered it in the form of published collections of its scripts, but then it turned out that my dad had some reel-to-reel tapes of it which he’d saved from his brief career in US radio. When I was a young teenager, the Goons were a private joke between myself and one of my friends, who’d discovered them independently.The show had a regular cast of three: creator and main writer Spike Milligan, actor and singer Harry Secombe, and Peter Sellers, who at the beginning of the show’s run was not famous and by the end was the most famous of them all. Milligan and Sellers played a variety of characters: Secombe usually played one, the ‘hero’, Ned Seagoon, a parody of a Heroic British Chap who was invariably portrayed as a bit of an idiot.Anyway, in one of the episodes, Seagoon is asked by another running character, Henry Crun (Sellers), an incredibly elderly man, to give his personal details. In order to imagine this joke, you have to imagine Sellers’ teeth-grindingly slow and quavery delivery as he laboriously repeats everything Secombe tells him:Henry: Let us get some details and documents...we must have the documents, you know. I'll just take a few particulars. Now, let's get the details and the documents...we must have the documents, you know.Seagoon: Of course.Henry:...Must have the documents. Ymnbnkhmn [mumbling], now, what's all this about? Oh yes, yes. Your name?Seagoon: Neddie Pugh Seagoon.Henry: N, E, D, D, I, E... Neddie — what was next?Seagoon: Neddie Pugh Seagoon.Henry: Pugh, P, H, E, W.Seagoon: No no no, it's pronounced "Phew" but it's spelt "Pug".Henry: P, U, G, H, P, U, G, H, yes, P, U, G, H, — there — Neddie Pugh, Sea-dune wasn't it?Seagoon: Yes...Seagoon, S, E, A, G, O, O, N.Henry: Could you spell it?Seagoon: Certainly — S, E, A, G, O, O, N.Henry: Seagoon... S, E, A, er — mnkk — mnkk [Snores]Seagoon: G, O, O, N, Seagoon.Henry:...Oh yes yes yes, good, good, yes, yes, yes, the full name. Now er — address?Seagoon: No fixed abode.Henry: No... F, I, X, E, D, fixed... A, B...Seagoon: A, B, O, D, E.Henry: O, D, E... There we are — No fixed abode — What number?Seagoon: 29A.Henry: 29A... Twenty-nine...A... District?Seagoon: London, SW 2.Henry: L, O, N, D, O, N - South West E, S, T... Two, wasn't it?Seagoon: Yes, two.Henry: T, W... It's no good, I'd better get a pencil and paper and write all this down.

Below is a basic explanation what is Zionism.It is a movement established to collect the Jewish people from the diaspora and bring them back to their indigenous, historical ancestral Homeland.The Zionist movement was founded by Theodor Herzl in 1897. However, the history of Zionism began earlier and is related to Judaism and Jewish history.The Hovevei Zion, (Lovers of Zion), were responsible for the creation of 20 new Jewish cities in the Southern Ottoman Sajaks (later Palestine) between 1870 and 1897.Before the Holocaust, the Zionist's movement central aims were the creation of a Jewish National Home and cultural center in then Palestine, by Facilitating Jewish migration.After the Holocaust, the movement focused on creating a "Jewish State" (a secular state with a Jewish majority) attaining its goal in 1948 with the creation of Israel.There Is no other example In human history of a "nation" being restored after such a long period of existence as a Diaspora.The success of Zionism shows that the percentage of the world's Jewish population who live in Israel has steadily grown over the years, and today 55% of the world's Jews live in Israel.While then Palestinian Arabs made efforts to curb the influx of Zionist emigration and lands purchases beginning in 1880, the recognition of Israel as an independent Jewish state after WWII, fulfilled the dream of statehood for thousands of years."Zionism is an indigenous people’s repatriation/liberation movement. It is thought to have officially emerged in exile, borne of the radical liberationist strain of Enlightenment thought from which feminism also emerged, but the underlying concept is much older. The yearning to return to our homeland has been ingrained in our culture ever since we were jettisoned from our soil by foreign occupiers, primarily into Europe, North Africa, and other parts of the Middle East.Zionist leaders sought the support of world powers, particularly the British (who would eventually betray the Zionists) and the King of Iraq, and began to make their way home – rebuilding their people and country into the powerhouses they are today.Zionism is, at its core, an indigenous rights project, and has been since day one. The Jews returning from exile had no mother country to “colonize” on behalf of. Israel IS the mother country. There was no New Warsaw, New Bielystok, New Vienna – only the revitalized names of ancient Jewish cities; cities that had been established by the ancestors of these supposed “European colonists”. The Zionists wanted nothing more than to rid themselves of diaspora and return home, and to compare that with colonialism is both dishonest and cruel."Leo Pinsker argued that while emancipation may solve the problems of individual Jews, it will not solve the problem of the Jewish nation. If the Jewish people do not acquire the effective external attributes of a nation, they will remain "everywhere as guests" and "nowhere at home."2 Assimilation would be "national death," whereas a Jewish state would not only provide the benefits of emancipation under the guise of "normalization," but a place of refuge where Jews could manage affairs in their own way without the perpetual stigma of minority status. With this reasoning, Zionism was as much an instance of nineteenth century nationalism as it was a response to emancipation and antisemitism (Avineri 1981, 13; Reinharz and Shapira 1995, 7).Let’s Talk About ColonialismHistory of Zionism - WikipediaIranian ZionismIranian Zionism with Jason Reza JorjaniCaroline Glick: A great—but fragile—triumph of ZionismAnalysis: Why Young Europeans Can’t Understand Israeli ‘Settlers’How a Former Non-Zionist Became a Supporter of Israel