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Adding vectors (2013)

Page history last edited by Joe Redish 6 years ago

Class Content I The Main Question: Motion > Kinematics

3.1.2.1

 

Prerequisites

 

We have described a mathematical structure that allows us to code both direction and magnitude of a quantity -- vectors. One way to think of it is as an arrow on a coordinate system starting from the origin and going off in a particular direction for a certain distance. The basic physical system we map onto this mathematical system is the displacement of an object in space.

 

A mental model for adding vectors

This physical model gives us a quick and easy way to think about how to add two vectors. You start at the origin and traverse the first vector from tail to head. (You undergo the first displacement.) Then, from where you are, you traverse the second arrow from tail to head. (You undergo the second displacement.) The result is the arrow drawn from the tail of the first arrow directly to the head of the second. (Your overall displacement.)

 

A graphical method for adding vectors

This sense of meaning of vector addition gives us a graphical way to add vectors. For example,  if we were consider displacement in a 2D space described by a "graph for the eye" that is an x-y plane, and we had two displacements labeled r1 and r2, then the sum of these two displacements, r, is what you get by doing one displacement after the other as shown in the figure:

 

  

This provides a graphical method for adding two vectors.

An algebraic way to add vectors

We also showed in our discussion of vectors how one could represent them algebraically using the unit direction vectors i and j. A general 2D vector can be represented by the sum of displacements in two perpendicular directions: x and y. An arbitrary vector (often written as (x,y) in math class) that has a displacement of the amount x in the x direction and an amount y in the y direction can be written

If we have two such vectors and want to add them, we can do so using straightforward algebra: rearranging and regrouping.

 

If we then identify the sum, r, as having components x and y, we can identify what they are:

So this gives the satisfying result that to add two vectors, we just add the x components and call it the new x component, and add the y components and call it the new y component.

 

You can see the relation of these in various forms of the rule that you may encounter.

 

Subtracting vectors

Subtracting vectors is no more difficult than adding vectors, since vector math follows the standard rules of algebra. Subtracting a vector is the same as adding the negative of the vector. (In standard algebra, -2 = +(-2).) For example

Geometrically, subtracting is the same as adding the reversed vector.

 

Follow-ons:

 

 Workout: Adding vectors

 

 

Joe Redish 10/4/13

 

 

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