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Problem 54 Hard Difficulty

If $ v_1, v_2 $ and $ v_3 $ are noncoplanar vectors, let
$ k_1 = \dfrac{v_2 \times v_3}{v_1 \cdot (v_2 \times v_3)} $ $ k_2 = \dfrac{v_3 \times v_1}{v_1 \cdot (v_2 \times v_3)} $
$ k_3 = \dfrac{v_1 \times v_2}{v_1 \cdot (v_2 \times v_3)} $
(These vectors occur in the study of crystallography. Vectors of the form $ n_1 v_1 + n_2 v_2 + n_3 v_3 $, where each $ n_i $ is an integer, form a lattice for a crystal. Vectors written similarly in terms of $ k_1, k_2 $, and $ k_3 $ form the reciprocal lattice.)
(a) Show that $ k_i $ is perpendicular to $ v_j $ if $ i \neq j $.
(b) Show that $ k_i \cdot v_i = 1 $ for $ i = 1, 2, 3 $.
(c) Show that $ k_1 \cdot (k_2 \times k_3) = \frac{1}{v_1 \cdot (v_2 \times v_3)} $.

Answer

a. $k_{1} \cdot v_{1}=\frac{\left(v_{2} \times v_{3}\right) \cdot v_{1}}{\left(v_{2} \times v_{3}\right) \cdot v_{1}}=1,$ and so on
b. $k_{1} \cdot v_{1}=\frac{\left(v_{2} \times v_{3}\right) \cdot v_{1}}{\left(v_{2} \times v_{3}\right) \cdot v_{1}}=1,$ and so on
c. $\mathbf{k}_{1} \cdot\left(\mathbf{k}_{2} \times \mathbf{k}_{3}\right)=\frac{1}{\mathbf{v}_{1} \cdot\left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)}$

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Video Transcript

Welcome back to the last problem of the cross product section. This time we're looking at three different vectors defined as follows up here, which are used in crystallography. This is a three part problem. So the first thing we want to do is verify that K one is perpendicular to V two and V three. And one of the ways we can do that is look at the dot product of K one and V two. If we write this out, we get K one, which is V to cross V three Divided by V one. Diet V to Cross V three. So this is K one started with vector V two. And since everything on the bottom here is just a scalar, this is the same thing as all of this started with V two. Now remember for across product V to cross V three is perpendicular to both V two and V three, This is perpendicular to V two, therefore something perpendicular to V two dotted with V two Has to be zero. Similarly, If we look at K one Dotted with V three. The same thing happens V to cross V three is perpendicular to V three And something perpendicular to V3.3 3 equals zero. We can do the same thing with K two. K two is V three cross V one divided by or to nominate her for you to cross. Absolutely. A little clear V to Cross V three. And if we dot this with the one, we have something perpendicular to V one dot V one, That's going to be zero. And if we got this with V three, We'll have something perpendicular to V three dotted with V three, That gives us zero. Let's do the last one really quick. K three is V one Cross v two divided by a denominator B1.5 to cross V three. Again, that's just a scalar. So if we got this with the one, you have something perpendicular to v. one V one that equals zero. And if we Takes about product with P two, We have something perpendicular to v. two. That'd be too, But also equal zero. Now the question is what happens if You look at K 1? Diet the one, well, let's make a little bit more room. K one dot V one. Let's look at K one again really quickly to cross v three. Okay, 1.5 1 is V to cross the three divided by our denominator. V one V 2. Cross v three. This is K one. And if we got this with V one, that's the same thing as our numerator. Started with V one or equivalently V one dot V to Cross V three divided by V one. Diet V to cross V three. And since our numerator and denominator are the same, that's just one. The same thing will happen if we dot K two with V two or K three with V three. We can see that really quickly up here. This dotted with V two by our triple product rule is equivalent to the denominator, So that equals one. And if we got this with V three by our triple product rule, that's equivalent to the denominator. And so that would equal one as well. Now where this gets really tricky is part three where it asks us what is K one dot K to Cross K three. So we've got K one Diet hey, to cross K three. It's one of the little trickier, let's remember that our denominators are just a scalar and so we'll end up just multiplying by one over that denominator twice. And that means what we're really looking at is K one Diet. And then I've taken the liberty of expanding out. V three. Cross v one Cross V one Cross v two. This entire mess here all divided by Uh huh. All divided by our denominator. The one giant. The two Cross V three times itself. So we can destroy view on dot be too crusty three squared. All right. Let's see if we can do this reasonably easily. K one dot. And let's start simplifying. Mm hmm. All right. So by our triple product rule V three cross V one dot V two is the same thing as V one dot V to Cross V three. All of that Times V one minus. And then we've seen this already. The three cross V one is perpendicular to V one and so something perpendicular to V one dot V one. It's just going to be zero V two, all divided by everyone. Diet V to Cross V three squared. And so you'll notice we've got one of these in the numerator, in two of these, in the denominator. What we're really looking at is let's bring all of these constants out front one over V one dot V to Cross V three, cancelling out once in the numerator, once in the denominator times K one dot the one. But we already know from above at K one dot V one equals one. Therefore, anyone got To cross K three is equal to one over V one dot V two cross V three. And that concludes the section. Thanks for watching.