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Problem 48 Medium Difficulty

If $ a + b + c = 0 $, show that
$$ a \times b = b \times c = c \times a $$

Answer

$a+b+c=0,$ thus $a=-b-c$

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Video Transcript

Welcome back to another cross product problem where we're trying to show if the sum of vectors A, B and C equals zero, that means a cross B equals B. Cross C equals C. Cross A. How do we get there? Well, if A plus B plus C equals zero, that tells us that A equals negative B minus C. Therefore a cross B is really negative B minus C. Crosby and cross products distribute. And so we have this is negative B. Crosby minus C. Crosby. But the cross product of any two parallel vectors, It's going to be zero minus. See Crosby. And another thing we know is that negative, be crusty? Is the same thing as positive be cross C. So there's our first equality. Next thing we can do is the same idea. But we'll start with writing B equals negative A minus C. And then we can look at B Cross C be crushed. C is then going to be negative A minus C. Chrissie. Again, this distributes, so we're looking at negative A cross C minus C cross C. Once again negative A. Craftsy. It's the same thing as C cross A. And any vector cross product with itself or any vector parallel to it zero. And therefore we have shown The 2nd equality as well. Thanks for watching.