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Old Apr 22nd 2018, 12:12 AM   #1
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Why is the product of a scalar and a vector, a vector?

Refer the title.
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Old Apr 22nd 2018, 02:00 AM   #2
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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?"
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Old Apr 23rd 2018, 02:11 AM   #3
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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

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Old Apr 23rd 2018, 04:23 AM   #4
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Originally Posted by avito009 View Post
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.

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Old Apr 23rd 2018, 04:27 AM   #5
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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.
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Old Apr 23rd 2018, 05:24 AM   #6
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Originally Posted by avito009 View Post
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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Old Apr 23rd 2018, 05:30 AM   #7
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Originally Posted by avito009 View Post
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 View Post
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.
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Old Apr 23rd 2018, 05:42 AM   #8
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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 <vector product>.
(Note that, unless the multiplying vector has a length of one, the magnitude will also change)

Watch out for the difference between the <vector product> of two vectors and the <scalar product> of two vectors.
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Old Apr 23rd 2018, 08:29 AM   #9
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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.
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Old Apr 23rd 2018, 03:28 PM   #10
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Originally Posted by avito009 View Post
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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