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I have read in couple of books that describe addition of vectors not acting at the same point. They describe it by saying we have to bring the initial point of one vector to the other and then resolve them into their respective x and y components and add the respective components to get the resultant.How is it that we can pull one vector to another point and still get the same effect?This seems physically impossible and very unintuitive. How is possible?

PanD
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  • We always assume (while doing mathematics) that when a vector is transmitted parallel to itself keeping its length and direction constant, it remains unchanged. – Qwerty Jun 19 '16 at 18:32
  • Isn't it a little weird physically? Let's assume the vector to be force a vector acting on a body. While moving it parallely if the vector comes out of the body on which it was initially acting, then it's effect itself becomes null. – PanD Jun 19 '16 at 18:35
  • I said , for mathematical calculations to be made easy, it is done. Physically point of application of forces is very important – Qwerty Jun 19 '16 at 18:45
  • So moving around vectors is just for mathematical convenience. And addition of vectors not acting at a single point has no physical significance. Kindly correct me if I'm wrong. – PanD Jun 19 '16 at 18:50
  • Somewhat .. for eg, often multiple vectors acting at distant places on the same body are brought together to calculate a resultant .(Not torque, of course!) .. – Qwerty Jun 19 '16 at 19:15

1 Answers1

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A finite directed line segment (in the plane, say, for simplicity) has a starting point, a direction (slope) and a length.

A vector is not a directed line segment.

Rather, a vector is the class of all the directed line segments having the same direction and length. Each individual directed line segment in the class is a representative of the vector.

In other words for any given vector $v$, there is a representative of $v$ at every point of the plane.

We are always free to choose the most convenient representative of a vector. If we want to add one vector to another, we are free to choose a representative of the first vector based at the origin; and a representative of the second vector based at the head (pointy end, if you think of it that way) of the first; and then draw the diagonal as a representative of the vector sum.

Given a vector, we are not "moving" it to a different location. The vector already has a representative at every point. All we are doing is choosing the most convenient representative for the given problem.

Also see Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$.

Community
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user4894
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  • If this vector is part of a vector field then can we place it anywhere in space according to our interest? – PanD Jun 19 '16 at 18:43
  • @PanD Not exactly, this is a little tricky. A vector field is an assignment of a vector to each point. But the vector itself is still the class of all directed line segments with given direction and length. So if you have a vector field, then at some point $p$ you assign a vector $v$. You can't just move that vector to some other point without possibly changing the vector field. The idea of a vector field is that there is some vector acting at that particular point. Even though the vector itself is an abstract direction and magnitude. – user4894 Jun 19 '16 at 18:56