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Problem 53 Hard Difficulty

Suppose that $ a \neq 0 $.
(a) If $ a \cdot b = a \cdot c $, does it follow that $ b = c $?
(b) If $ a \times b = a \times c $, does it follow that $ b = c $?
(c) If $ a \cdot b = a \cdot c $ and $ a \times b = a \times c $, does it follow that $ b = c $?

Answer

(a) No
(b) No
(c) Yes

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Video Transcript

Welcome back to another dot and cross products. Problem. Yeah. In this case we're trying to figure out if a dot B equals dot C, But a is not equal to zero. Does that mean at B is equal to see? Well, the answer actually is no. And we can see this in a couple different ways. First, let's imagine A B is equal to zero. That means that A and be are orthogonal, but a C is another vector orthogonal to a. Then a dot C is also going to be zero. And so we could have, the left side is equal to the right side. Even though B and C are clearly different vectors, we can see this a little more mathematically if we rearrange this equation. So it says a dot be minus a dot C equals zero. The mattress tells us that a diet b minus C equals zero and therefore B minus c. Just has to be perpendicular to A but it doesn't say that B has to equal. See. All right, what about if a cross B equals a cross C. Well, same idea. Yes. Let's say both sides are equal to zero again, that means that A and B are parallel and the D n C are also parallel. We had A B and see a Crosby would be zero. A Cross C would be zero, but B and C are very clearly different. Once again, we can see this by looking at a cross, B minus A cross C equals zero. And we know that since this distributes as well we have a cross B minus C equals zero, which just tells us that B minus C has to be parallel to A. But what happens if both this is true, A dot B equals C and this is true, A Crosby equals a cross C. What does that tell us about BNC this time? Well, just as before, we require B minus C to be perpendicular to A. But we also require B minus C to be parallel to A. And the only vector that's both perpendicular and parallel to another vector has to be the zero vector. Therefore b minus C has to be the zero vector, or equivalently B has to equal. See Thank.