State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar.
(a) $ a \cdot (b \times c) $
(b) $ a \times (b \cdot c) $
(c) $ a \times (b \times c) $
(d) $ a \cdot (b \cdot c) $
(e) $ (a \cdot b) \times (c \cdot d) $
(f) $ (a \times b) \cdot (c \times d) $
All right. So we're going to figure out whether these expressions are meaningful and if so whether they're scholars are vectors and I guess we're assuming the A. B, C. And D. Are all vectors. It's better if you um put a vector symbol when your handwriting. Um I didn't do it here because the question didn't have it. But I do think that's less confusing. Um That way you can tell in your work whether your values are vectors are scholars, but for now there's no vector symbols. Let's assume they're all vectors. So what happens? Um Well what you need to know about across product is that it's a vector crossing a vector that will give you a vector. And what you need to know about adopt product is that it's a vector dotted with a vector. That will give you a scaler. So now all we have to do is see whether these makes sense. So um we have a vector cross. The vector. That's good. That'll give you a vector. This whole thing is a vector and then a is a vector. So we have a vector dotted with a vector. This seems to make sense. Um And then the answer to adopt products should be a scaler. Okay, what about B? We have a vector dotted with a vector. That should give us a scaler. But now we're trying to do a vector crossed with a scaler. That doesn't make sense. That's a meaningless statement. Um So that's what it means when the same meaningless, something like that for Patsy. We have always do the parentheses. First we have a vector dot crossed with the vector we get a vector. That's good. And now we have a vector crossed with the vector. That's good. We should get a vector for part D. We have a vector dotted with a vector. We get a scaler and then we're trying to do a vector dotted with a scaler. That's no good. That's not normal multiplication. Um That's that's the type of multiplication that doesn't exist. Okay for part E. We have vector dotted with a vector is a scalar. I'm just going to write an s to remember vector dotted with a vector is a scalar. But now I'm trying to do scale across scaler. That doesn't work. That's meaningless. And for part F I have vector crossed with the vector. I get a vector vector crossed with the vector, I get a vector and now I have a vector dotted with a vector. That's good. I should get a scalar.