4.3: Geometric Meaning of Vector Addition
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Learning Objectives
- Understand vector addition, geometrically.
Recall that an element of \(\mathbb{R}^{n}\) is an ordered list of numbers. For the specific case of \(n=2,3\) this can be used to determine a point in two or three dimensional space. This point is specified relative to some coordinate axes.
Consider the case \(n=3\). Recall that taking a vector and moving it around without changing its length or direction does not change the vector. This is important in the geometric representation of vector addition.
Suppose we have two vectors, \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^{3}\). Each of these can be drawn geometrically by placing the tail of each vector at \(0\) and its point at \(\left( u_{1}, u_{2}, u_{3}\right)\) and \(\left( v_{1}, v_{2}, v_{3}\right)\) respectively. Suppose we slide the vector \(\vec{v}\) so that its tail sits at the point of \(\vec{u}\). We know that this does not change the vector \(\vec{v}\). Now, draw a new vector from the tail of \(\vec{u}\) to the point of \(\vec{v}\). This vector is \(\vec{u}+\vec{v}\).
The geometric significance of vector addition in \(\mathbb{R}^n\) for any \(n\) is given in the following definition.
Definition \(\PageIndex{1}\): Geometry of Vector Addition
Let \(\vec{u}\) and \(\vec{v}\) be two vectors. Slide \(\vec{v}\) so that the tail of \(\vec{v}\) is on the point of \(\vec{u}\). Then draw the arrow which goes from the tail of \(\vec{u}\) to the point of \(\vec{v}\). This arrow represents the vector \(\vec{u}+\vec{v}\).
This definition is illustrated in the following picture in which \(\vec{u}+\vec{v}\) is shown for the special case \(n=3\).
Notice the parallelogram created by \(\vec{u}\) and \(\vec{v}\) in the above diagram. Then \(\vec{u} + \vec{v}\) is the directed diagonal of the parallelogram determined by the two vectors \(\vec{u}\) and \(\vec{v}\).
When you have a vector \(\vec{v}\), its additive inverse \(-\vec{v}\) will be the vector which has the same magnitude as \(\vec{v}\) but the opposite direction. When one writes \(\vec{u}-\vec{v,}\) the meaning is \(\vec{u} + \left( -\vec{v}\right)\) as with real numbers. The following example illustrates these definitions and conventions.
Example \(\PageIndex{1}\): Graphing Vector Addition
Consider the following picture of vectors \(\vec{u}\) and \(\vec{v}\).
Sketch a picture of \(\vec{u}+\vec{v},\vec{u}-\vec{v}.\)
Solution
We will first sketch \(\vec{u}+\vec{v}.\) Begin by drawing \(\vec{u}\) and then at the point of \(\vec{u}\), place the tail of \(\vec{v}\) as shown. Then \(\vec{u}+\vec{v}\) is the vector which results from drawing a vector from the tail of \(\vec{u}\) to the tip of \(\vec{v}\).
Next consider \(\vec{u}-\vec{v}.\) This means \(\vec{u}+\left( -\vec{v} \right) .\) From the above geometric description of vector addition, \(-\vec{v}\) is the vector which has the same length but which points in the opposite direction to \(\vec{v}\). Here is a picture.