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Problem 51 Medium Difficulty

Use Exercise 50 to prove that
$$ a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0 $$

Answer

$a \times(b \times c)=(a \cdot c) b-(a \cdot b) c$

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Video Transcript

Yes, welcome back to another cross product problem. Our last proof, we showed that across peak rossi is equal to a dot C times b minus a dot B times C. This time we're going to use that to evaluate a cross the cross C plus B. Cross secrecy plus C. Cross a Crosby and see what that's all equal to. So, using this identity, we can start to evaluate these terms by term by term a crispy crust. See, has already written on the page, it's a dot C times b minus a dot be time see plus. Then we're looking at B cross, see, cross A. Looking at the first in third terms here, that will be be dot A time see minus. And then the 1st and 2nd terms here, that will be be dot C times A plus. And then same idea. We're looking at the First and 3rd terms in the dark product will be c dot be times a minus. And in the 1st and 2nd terms c dot A times B. Yeah. Well, notice that we've got an A dot C, times B and a minus C dot A times B since a dot C and see that they are the same thing those are going to cancel. We've also got a B dot A times C right here and we've got a negative a tubby time. See right here again, A dot B and P dot A are the same things. Those canceled. And lastly, sea dot B times a negative beat at C times A. There's going to cancel as well. And we're left with A big old zero. The sum of all of these triple cross products is just zero. All right. Thank you for watching.