I have a question in that if v is s any nonzero vector, and v is...

I have a question in that if v is s any nonzero vector, and v is positioned with its initial point at the origin, then the terminal points of all scalar multiples of v will occur at all the points on a straight line through the origin. But if we want to find two vectors, let's say m and n, that are parallel to each other, we need to determine whether they are multiples of each other.

So my question here is: what's the difference between scalar multiples and multiples? In another word, vectors within the same line are parallel to each other because this would be a special form of parallel?

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Expert Solution

If m and n are two vectors and if 'a' is any scalar then m and n are scalar multiples of each other means

m=(a) (n) =an.

Here a(n) is a vector parellel to the vector m whose direction is same as vector m and its magnitude is 'a' times the magnitude of m.

so the scalar multiple always represent a parellel vector or as a special case collinear vector.

But if we take only multiple of each other then it gives (m) (n).

means multiplication of two vectors which is not correct.

Multiplication of two vectors means it should be a dot product or cross product.

Therefore multiples of each other is entirely different from scalar multiple.

For example vector 2 means it is a scalar multiple of . Here 2 is another vector say whose direction is same as (parellel to ) and magnitude is two times the magnitude of .

Also - 2 is a vector whose direction is opposite of but magnitude is 2 times the magnitude of .

milcah answered 1 year ago

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