Vector Addition

Vector Addition

We now proceed to define addition of vectors in R^2, and multiplication of vectors in R^2by real numbers. In this context, we interpret R^2 as a set of vectors represented by arrowswith tail at the origin.

Given two vectors a=(a_1,a_2) and b=(b_1,b_2) in R^2, we use a+b to denote the vector (a_1+b_1,a_2+b_2) in R^2 obtained by adding the components separately. We call a+b the sum of a and b obtained through vector addition. Thus if  a=(a_1,a_2)  and  b=(b_1,b_2) are given vectors in R^2, then

  • (a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2+b_2),

which says that vector addition is carried out by adding components separately. We note that a+b=b+a since a_1+b_1=b_1+a_1 and a_2+b_2=b_2+a_2. We say that 0=(0,0) is the zero vector since a+0=0+a=a for any vector a. Note the difference between the vector zero and its two zero components, which are usually scalars

Example: We have (2,5)+(7,1)=(9,6) and (2.1,5.3)+(7.6,1.9) =(9.7,7.2).

Vector Addition and the Parallelogram Law

We may represent vector addition geometrically using the Parallelogram Law as follows. The vector a+b corresponds to the arrow along the diagonal in the parallelogram with two sides formed by the arrows a and b:

This follows by noting that the coordinates of the head of a+b is obtained by adding the coordinates of the points a and b separately, as just illustrated.

This definition of vector addition implies that we may reach the point (a_1+b_1,a_2+b_2) by walking along arrows in two different ways.

First, we simply follow the arrow (a_1+b_1,a_2+b_2) to its head, corresponding to walking along the diagonal of the parallelogram formed by a and b.

Secondly, we could follow the arrow a from the origin to its head at the point (a_1,a_2) and then continue to the head of the arrow \bar b parallel to $b$ and of equal length as b with tail at (a_1,a_2).

Alternative, we may follow the arrow b from the origin to its head at the point (b_1,b_2) and then continue to the head of the arrow \bar a parallel to a and of equal length as a with tail at (b_1,b_2).

The three different routes to the point latex (a_1+b_1,a_2+b_2) are displayed in the above figure.

We sum up in the following theorem:

Theorem: Adding two vectors a=(a_1,a_2) and b=(b_1,b_2)  in R^2 to get the sum a+b=(a_1+b_1,a_2+b_2)  corresponds to adding the arrows a and b using the Parallelogram Law.

In particular, we can write a vector as the sum of its components in the coordinate directions as follows:

  • (a_1,a_2)=(a_1,0)+(0,a_2),
as illustrated in the following figure:
A vector a represented as the sum of two vectors parallel with the coordinate axes.

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