Vectors in the plane

Our ad hoc definition of a plane vector is that it is a pair of numbers written vertically in an array. The idea came from listing coefficients of a variable in a pair of equations. Indeed, with this idea, it is easy to imagine a vector of any height, since we can easily imagine many equations involving the same variables.

Our original equations came out as setting a combination of vectors equal to a given vector. For example:
Given equations: , we wrote them as
and this was expressing the vector as a linear combination of the vectors and with coefficients respectively.

But we have seen or heard the term in other contexts and wish to see the connection. Here is the first geometric meaning for vectors of height two like this.

Consider a point with coordinates ( ) and associate the vector to it. Geometrically, we think of the vector as a directed line segment from the origin O( ) to the point P( ).

In general, let us say that the position vector of a point is simply the vector and conversely we say the point is its tip.

At least this sets up a nice correspondence and we now investigate what meaning can the algebraic operations have.

The operation of multiplying the vector by some number, say causes the line segment OP to become three times as long. If we use the multiple by -3, then it has the same change in length, but it also flips about the origin. In general, we define a scalar multiple of a vector by a constant (or scalar) x to be a new vector . This causes a scaling in length by a factor of with a flip involved, if is negative.

Observe the pictures below.

> u:=DV([0,0],[1,1]):
v:=DV([0,0],[1,-1]):
u1:=DV([0,0],[3,3],``,cyan):
v1:=DV([0,0],[-2,2],``,cyan):
w:=DV([0,0],[1,5],``,green):

> display([u,v,u1,v1,w,
PT([1,-1],` v`,20),PT([1,1],` u`,20),PT([3,3],` 3u`,20)
,PT([-2,2],` -2v`,20),PT([1,5],` 3u-2v`,20)]);

> u:='u':v:='v':

>

We can also add vectors by adding the corresponding components and we have indeed computed some vectors above. Note for instance , and their sum . It can be easily seen their tips together with the origin form a parallelogram and this illustrates the so-called parallelogram law of addition.