Physics Help Forum Why is the product of a scalar and a vector, a vector?
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 Apr 22nd 2018, 12:12 AM #1 Senior Member     Join Date: Feb 2017 Posts: 202 Why is the product of a scalar and a vector, a vector? Refer the title.
 Apr 22nd 2018, 02:00 AM #2 Senior Member   Join Date: Aug 2010 Posts: 347 The first thing one would need to know, to answer the general question "Why is A a B" is the definition of "A"! You are asking "why is the product of a scalar and a vector a vector" so I would ask "what is the definition of the product of a scalar and a vector?" benit13 likes this.
 Apr 23rd 2018, 02:11 AM #3 Senior Member   Join Date: Oct 2017 Location: Glasgow Posts: 207 As the operations are currently defined for vectors, scalar multiplication is distributive, similar to the following algebra: $\displaystyle 5(x + y) = 5x + 5y$ Similarly: $\displaystyle 5(\hat{i} + \hat{j}) = 5\hat{i} + 5 \hat{j}$ A scalar changes the magnitude of the vector, but not its direction if the coordinate system has a set of orthogonal basis vectors Last edited by benit13; Apr 23rd 2018 at 02:14 AM.
Apr 23rd 2018, 04:23 AM   #4
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 Originally Posted by avito009 Refer the title.
It seems to me that I always do my very best thinking when I'm on "the throne."

What does multiplication by a number mean to you? S'pose I wrote 3 x 5 dollars. I'm sure you'd know the answer to that

3x5 dollars = 5 dollars + 5 dollars + 5 dollars = 15 dollars

. Extend that to vectors. What should 5 x A = 5A mean? It means

5A = A + A + A + A + A

In other words you're taking a vector and multiplying it to get a larger or smaller vector in the same direction.

Last edited by topsquark; Apr 23rd 2018 at 05:18 AM.

 Apr 23rd 2018, 04:27 AM #5 Senior Member     Join Date: Feb 2017 Posts: 202 Eucledian space. Now if we take magnitude of a vector it is >=0. Which menas magnitude cant be negative. So a scalar is a quantity having only magnitude. Which means its positive. This holds only in Euclidean space. So the line in x axis where -1 and + 2 is taken if -1 is taken to be a vector then it signifies 1 in the backward direction and +2 means 2 in forward direction. So here -1 is a vector and |1| is a scalar. Since scalar cant have a negative direction. But in case of vector space. What study is vector space? Its linear algebra. So in linear algebra a scalar can have a negative value. Please correct me if I am wrong.
Apr 23rd 2018, 05:24 AM   #6

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 Originally Posted by avito009 Now if we take magnitude of a vector it is >=0. Which menas magnitude cant be negative. So a scalar is a quantity having only magnitude. Which means its positive. This holds only in Euclidean space. So the line in x axis where -1 and + 2 is taken if -1 is taken to be a vector then it signifies 1 in the backward direction and +2 means 2 in forward direction. So here -1 is a vector and |1| is a scalar. Since scalar cant have a negative direction. But in case of vector space. What study is vector space? Its linear algebra. So in linear algebra a scalar can have a negative value. Please correct me if I am wrong.
It all depends on where you are getting your scalars from. If we take the scalar to be a real number then we find that it can be negative, zero, or positive. But we can also take that scalar to be a complex number, and complex numbers are not positive or negative though they can, of course, be zero. The point is that the space of scalar values we are working with does not have to be related to the vector space in question.

This is all much more general than you are looking at. Given any field F and any vector space V we can construct an "F-vector space." In the example you are talking about F is a real number and V is the "usual" vector space on the plane.

-Dan
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Apr 23rd 2018, 05:30 AM   #7
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 Originally Posted by avito009 Now if we take magnitude of a vector it is >=0. Which menas magnitude cant be negative. So a scalar is a quantity having only magnitude. Which means its positive.
A scalar can be negative. All the negative sign means is that the new vector points in the opposite direction as the original.

And i you ever study relativity you'll see vectors whose magnitude itself is negative.

 Originally Posted by avito009 But in case of vector space. What study is vector space? Its linear algebra. So in linear algebra a scalar can have a negative value.
Although the elements of a vector space are called "vectors" in actuality the can mean a lot of things such as matrices, quantum states, etc.

 Apr 23rd 2018, 05:42 AM #8 Senior Member     Join Date: Jun 2016 Location: England Posts: 586 A vector has magnitude and direction. A scalar has only magnitude. You can change the magnitude of a Vector by multiplying it by a Scalar. You can change the Direction of a Vector by multiplying it by another Vector . (Note that, unless the multiplying vector has a length of one, the magnitude will also change) Watch out for the difference between the of two vectors and the of two vectors. __________________ ~\o/~
 Apr 23rd 2018, 08:29 AM #9 Senior Member     Join Date: Feb 2017 Posts: 202 Clarification. Let me clarify why I have doubt that a scalar cant be negative. One of the definitions of scalar would be this: Lets say we have an acceleration of -5 m/s (Deceleration actually). Now whats its scalar? Its |5| which is 5. Notice that the negative sign is dropped. So this made me think that a scalar is always positive in Euclidean space.
Apr 23rd 2018, 03:28 PM   #10

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 Originally Posted by avito009 Let me clarify why I have doubt that a scalar cant be negative. One of the definitions of scalar would be this: Lets say we have an acceleration of -5 m/s (Deceleration actually). Now whats its scalar? Its |5| which is 5. Notice that the negative sign is dropped. So this made me think that a scalar is always positive in Euclidean space.
You are making a number of threads which can make it hard to collate everything.

A vector cannot be positive or negative... It is not a number. If there is a negative involved then it must be from the scalar.

Let's get out of 1-D for a moment. Consider the vector $\displaystyle \textbf{v} = 5 \hat{i} - 2 \hat{j} + 3 \hat{k}$. Is this positive or negative? There is nothing to determine it one way or another so we can't say that it is either.

By the way, you really need to work on your units. m/s is the unit for a speed, m/s^2 is the unit of an acceleration.

-Dan

PS I'm sorry, I keep finding things to add. You say that your vector -5 has a scalar |-5| = 5. This is the length or size or more properly "norm" of the vector. It is a scalar but a vector does not have "a scalar" in this sense. There can be many properties of a vector that are real numbers.
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Last edited by topsquark; Apr 23rd 2018 at 03:32 PM.

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