Physics Help Forum Can only vectors be multiplied in Euclidean Space?
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 Apr 23rd 2018, 04:52 AM #1 Senior Member     Join Date: Feb 2017 Posts: 196 Can only vectors be multiplied in Euclidean Space? After googling I could see vector product and scalar product in Euclidean Space. But can scalar be multiplied by vector in Euclidean space? If it can then I have a conclusion from the previous post. If so we can conclude that the direction does not change when a scalar is multiplied by vector in Euclidean space because the scalar is either positive or 0. Scalar is nothing but the magnitude or length or norm of a vector.
 Apr 23rd 2018, 05:01 AM #2 Senior Member     Join Date: Feb 2017 Posts: 196 Conclusion. Lets say a vector 5 miles North is to be multiplied to a scalar 5 kilos. Then in Euclidean space you take the scalar |5| miles and multiply it with |5| kilos which gives |25|. Now what I am saying is that the scalar quantity from a vector can be multiplied to a scalar. The end product is 25 kilogram miles per second, North. Observe direction has not changed in Euclidean space. But direction can change in vector space. Is this correct?
Apr 23rd 2018, 05:12 AM   #3
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 Originally Posted by avito009 After googling I could see vector product and scalar product in Euclidean Space.
You need to be careful with your terms. The vector product and scalar product of two vectors in Euclidean space are also commonly known as the cross product and dot product.

What you are trying to look at is if we have a scalar a belonging to the complex number system and a vector v belonging to the "usual" vector space on the Euclidean plane, then can we multiply the two? If V is a vector space and $\displaystyle \textbf{v} \in V$ and $\displaystyle a \in \mathbb{C}$ then v' = av implies that $\displaystyle \textbf{v}' \in V$ as well.

Note, though, that a scalar can change the direction of a vector... If that scalar is a negative number then the direction points the other way.

-Dan
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Apr 23rd 2018, 05:16 AM   #4
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 Originally Posted by avito009 Lets say a vector 5 miles North is to be multiplied to a scalar 5 kilos. Then in Euclidean space you take the scalar |5| miles and multiply it with |5| kilos which gives |25|. Now what I am saying is that the scalar quantity from a vector can be multiplied to a scalar. The end product is 25 kilogram miles per second, North. Observe direction has not changed in Euclidean space. But direction can change in vector space. Is this correct?
More or less, though, check your units. When you multiply the scalar 5 kg by the vector 5 miles N you get 25 kg miles N, not kg m/s.

-Dan
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Apr 23rd 2018, 05:31 AM   #5
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 Originally Posted by topsquark Note, though, that a scalar can change the direction of a vector... If that scalar is a negative number then the direction points the other way. -Dan
My point is that in a euclidean space -10 would be a vector since it has both direction and magnitude. So a scalar cant be negative in Euclidean space because if it was so then it would be a vector.

Apr 23rd 2018, 06:25 AM   #6
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 Originally Posted by avito009 My point is that in a euclidean space -10 would be a vector since it has both direction and magnitude. So a scalar cant be negative in Euclidean space because if it was so then it would be a vector.
You have to specify a direction. -10 makes no sense as a vector except, perhaps, when considering a 1-D space. But even here -10 really represents the scalar -1 multiplied by some vector 10. As I said before, a scalar can be a negative number.

-Dan
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Apr 23rd 2018, 09:13 AM   #7
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 Originally Posted by avito009 My point is that in a euclidean space -10 would be a vector since it has both direction and magnitude.
-10 on its own is a scalar. $\displaystyle -10 \hat{i}$ is a vector, where the $\displaystyle \hat{i}$ is a base unit vector of the Cartesian coordinate system, which can be seen as an indication of the direction the vector is pointing.

Apr 23rd 2018, 03:00 PM   #8
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 Originally Posted by avito009 My point is that in a euclidean space -10 would be a vector since it has both direction and magnitude. So a scalar cant be negative in Euclidean space because if it was so then it would be a vector.
If it helps many Physics professors do not do a good job at explaining Physics is 1-D space. What they are trying to describe is a vector space (usually using the displacement) along a line. The - signs are actually indicators of direction, not value. A vector -10 m means, as benit13 says, is really $\displaystyle -10 \hat{i}$ indicating 10 m in the $\displaystyle -\hat{i}$ direction. But it's just easier for them to ignore the subtleties and call it -10 m. In fact the four motion in 1-D with constant acceleration equations are actually vector equations, just like their two and three dimensional analogues.

-Dan
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