Vectors

40.4. Geometric inter pretation of vector addition and multiplication

Suppose you have two vectors and . Consider them as position vectors, i.e. represent them by
vectors that have the origin as initial point:

Then the origin and the three endpoints of the vectors and form a parallelogram. See figure
15.

To multiply a vector with a real number t you multiply its length with |t|; if t < 0 you reverse the
direction of .

Figure 15. Two ways of adding plane vectors, and an addition of space vectors

40.9 Example. In example 40.5 we as sumed two vectors and were given, and then defined
and . In figure 17 the vectors and are constructed geometrically from
some arbitrarily chosen and . We also found algebraically in example 40.5 that .
The third drawing in figure 17 illustrates this.

Figure 16. Multiples of a vector, and the difference of two vectors .

Figure 17. Picture proof that in example 40.9.

41. Parametric equations for lines and planes

Given two distinct points A and B we consider the line segment AB. If X is any given point on AB then we will now find a formula for the position vector of X. Define t to be the ratio between the lengths of the line segments AX and AB, Then the vectors   and are related by . Since AX is shorter than AB we have 0 < t < 1. The position vector of the point X on the line segment AB is If we write for the position vectors of A, B, X, then we get This equation is called the parametric equation for the line through A and B. In our derivation the parameter t satisfied 0 ≤ t ≤ 1, but there is nothing that keeps us from substituting negative values of t, or numbers t > 1 in (53). The resulting vectors are position vectors of points X which lie on the line through A and B. 41.1 Find the parametric equation for the line through the points A(2, 1) and B(3,−1), and de termine where intersects the axis. Solution : The position vectors of A, B are and , so the position vector of an arbitrary point on is given by Figure 18. Constructing points on the line through A and B

where t is an arbitrary real number. This vector points to the point X = (1+ 2t, 2 − 3t). By definition, a point lies on the -axis if its comp onent vanishes . Thus if the point X = (1 +2t, 2 − 3t) lies on the x1-axis, then 2− 3t = 0, i.e. . The intersection point X of and the -axis is therefore 41.2 Midpoint of a line segment. If M is the midpoint of the line segment AB, then the vectors and are both parallel and have the same direction and length (namely, half the length of the line segment AB). Hence they are equal: . If , and are the position vectors of A , M and B, then this means Add and to both sides, and divide by 2 to get

41.1. Parametric equations for planes in space*

You can specify a plane in three dimensional space by naming a point A on the plane P, and two
vectors and parallel to the plane P, but not parallel to each other. Then any point on the plane P
has position vector given by

The fol lowing construction explains why (54) will give you any point on the plane through A,
parallel to

Let A , be given, and suppose we want to express the position vector of some other point X on
the plane P in terms of and .

First we note that

Next, you draw a paralle logram in the plane P whose sides are parallel to the vectors and , and
whose diagonal is the line segment AX. The sides of this parallelogram represent vectors which are
multiples of and and which add up to . So, if one side of the parallelogram is s and the other
t then we have . With   this implies (54).

Figure 19. Generating points on a plane P

42. Vector Bases

42.1. The Standard Basis Vectors

The notation for vectors which we have been using so far is not the most traditional. In the late
19th century GIBBS and HEAVYSIDE adapted HAMILTON’s theory of Quaternions to deal with vectors.
Their notation is still popular in texts on electromagnetism and fluid mechanics.

Define the following three vectors:

Then every vector can be written as a linear combination of and namely as follows:

Moreover, there is only one way to write a given vector as a linear combination of . This means that

For plane vectors one defines

and just as for three dimensional vectors one can write every (plane) vector as a linear combination
of and ,

Just as for space vectors, there is only one way to write a given vector as a linear combination of and
.