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Prove directly (no CP or IP)1 (AxJAx3 (xJ(Bx 3 Cx) 2. Am Bm ICmHTML Editonda B I 4 A - @ - I E 2992 X X EB- 0 & & 0 v % & v D] M # 12pt...

Question

Prove directly (no CP or IP)1 (AxJAx3 (xJ(Bx 3 Cx) 2. Am Bm ICmHTML Editonda B I 4 A - @ - I E 2992 X X EB- 0 & & 0 v % & v D] M # 12pt

Prove directly (no CP or IP) 1 (AxJAx3 (xJ(Bx 3 Cx) 2. Am Bm ICm HTML Editonda B I 4 A - @ - I E 2992 X X E B- 0 & & 0 v % & v D] M # 12pt



Answers

Prove the following identities. Assume $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and $\mathbf{x}$ are nonzero vectors in $\mathbb{R}^{3}$. $$(\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{x})=(\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{x})-(\mathbf{u} \cdot \mathbf{x})(\mathbf{v} \cdot \mathbf{w})$$

The following problem, we want to show that y double prime equals and negative A C Times M -1 X. to the M -2 over b squared. Um Why To the to M -1 given the following information. So what this is going to look like is if we have A X. But I am Mhm. That's me X. But why B Y to the B Y to them? Yeah. See mm. What we can do is subtract this X e m component. We can divide by B. And we can take the M through of this function. So we take the M through, put it to the one over M power. When we do so we can now take the derivative and then the second derivative and we'll find that it equals what we're looking for.

As we all know that we have an A Z. Of a particle when the particle is moving with relativistic speed is equal to gamma multiplication and not be noticed arrest an easy so gamma is equal to E by in a Also we can write the value of gamma edge won by 100 would one minus B squared by c square. So from both the expressions I can write the expression is one by understood. One minus P square by C square is equal to E by the note. On further simplification I can died under route one minus B squared by C square is required to denote by E squiring both side. I can't id the expression else one minor P squared by C square is equal to a note by E. Holy square solving it for the B. Y. C is equal to one minus he. Not by E. Holy square hold to the power one by two. Using the binomial Thuram, I can right from the value of the way I see. Using the binomial expansion, I can write it as one minus one by two multiplication, entered by E. Holy Square W as the value of each very greater than the note. So we can neglect higher terms. And the value of B, Y. C can be written as one minus one by two multiplication in order by E fully square, so it is horrified.

Either. Today we're going to prove the statement that you cross the dot w cross X is equal Teoh you dot w times vida x subtract you don't x times If you don't. W we're going to do this by using u equals u one I plus U two j plus you three k on similar statements for V, W and X when we're gonna show that the left hand side is equal to the right hand side. So, uh, you cross the by the determinant method we have high J k. You want you to you three The one v two b three This is equal toe I times you to you three v to v three Uh minus j times the determinant You one you three of the u R v three plus k times a determined you want you to Do you want me to, Which is equality, I times um you two v three minus you three v two plus j times the one you three minus v three u one plus k times the determinant. You want to be too minus you to the one on by substituting and w for you on day X for V. We can see that doubIe cross X is equal to i Times w two x three minus W three x two plus j at times x one w three minus X ray w one plus k times uh, w one x two minus two You two x one, then u cross v dot w Term cross eggs is we multiply the I components. So this is, uh, you to be three. Fine issue three v two times W two x three minus debut three x two Uh, plus the J component. Ah, I got to swap these around so we have a uniform order you three v one minus you want V three times w three x one minus Toby You want extra, then Similarly for case component you want to be to minus YouTube V one W while x two minus to B two x one. This is equal to, um, it's expand. We'll get four terms per, um, line. Say this is equal to you to the three uh, W two x three plus you three. V two, w three x two Um, minus you three v two W two x three minus you to v three w three x two and Alex move on to, uh, second line plus you. Three v one w three x one Um, plus, you won the three w one x three minus you three V one W three x one minus. You want the three? Ah W three. Excellent that I find online. Plus you want v two W one x two plus you to the one W two x one. Uh, minus you too. The one to be one x two uh minus. You want the to? So you two on day X one. So this gives us, huh? 12 components 12 01234567 812 Terms in this some, sir, is the left hand side. No need to calculate the right hand side which we shall separate off here. So, um, when in any dot product just as an aside, whenever we have w dot you don't love you. This just, uh you wonder you want less you to w two etcetera. Eso they're very simple to right down. So I don't need to pre calculate that we could just write them straight away. So this is equal to you want? W one plus you to w two plus you three w three times the while x one plus v two x two plus the three x three minus you one x one plus you two x two plus you three x ray times v one w one plus v two w two plus v three w three. So this is equal to, um, I'll first right out all of the terms that with same index, because this will be useful for later. You want w one plus YouTube w potus You want to be you on the wall x one plus you to W to be two x two plus you three w three v three X ray Now the rest of atoms you want w one v two x two Uh, plus you want to be one V three X ray plus you to to be you to V one x one plus ah, you are plus you to to be you two v three x ray plus you three Ah w three v one x one plus you www three v two x two. We're going to subtract again. We're going to have you all X one. The one w one plus you two x two v two w two plus you three x ray 33 w three. So these will or cancel with their first terms here, Then we have the rest of our terms. So we have subtract you want x one V two to be to minus you one x one e three w three unless you two x two v one w one minus you two x two v three w three minus you three x three v one w one minus you three x three The two w two. Now, if we have all of the same terms on the left and right hand side, then we will have shown that they're equal. So let's look for, uh, you to w two v three x three. Is this in positive? Selby on the, uh top. So it's Ah, you Teoh W two v three x ray. So that is here you www three v two and then we're looking for this term. Yeah, which is you? Three homes by council. The first you three, uh, W three b two x two, which is here now. We found that now onto subtractions. So, uh, you three, uh, horsey ordering you three x ray V two W two is just here. And then you, uh, you two x two, Uh, be three w three. It's just here. Then on pluses again, we have you three w three v one Exxon, which is here then. Ah, you one w on V three x ray, which is just here, then subtractions again, you three. Um w ah, three m Ah, a POTUS. Um, there must have been, um, and era when, uh, writing. Ah, this time out. Um, let me just check the two subtractions. Uh, you three, the wall that this term should have. Bean Uh ah. W one x three boats. So you three X ray W one V one. Um, is here. So that's that, Tim. And then, um you want X one V three, w three? Uh, there should be no one, but that's that to him. When additions you want w one v two x two, which is this term is just here. You to w to view on X one is just here then you two x two Ah, view on w one is just here on. Do you want X one of e T W two is just here, so the left hand side is equal to the right hand side, said a statement holds. So we've proven it, I think.

We know that angle views equivalent to any movie. They're common angles there for by the S a passer siding beside partial it. We know that triangle v w z a similar trying with the X y there for Antoine equals angle too. And w z is parallel to X y remember VX, a trans versatile from the corresponding angle property.


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