Vectors

45.1 Example. Let the points A(1, 0, 2), B(2, 0, 0), C(3, 1,−1) and D(2, 1, 1) be given.

Show that ABCD is a parallelogram, and compute its area.

Solution : ABCD will be a paralle logram if and only if . In terms of the position
vectors and of A, B, C, D this boils down to

For our points we get

So ABCD is indeed a parallelogram. Its area is the length of

So the area of ABCD is

45.2. Finding the normal to a plane

If you know two vectors and which are parallel to a given plane P but not parallel to each other, then you can find a normal vector for the plane P by computing We have just seen that the vector must be perpendicular to both and, and hence it is perpendicular to the plane P.

This trick is especially useful when you have three points A, B and C, and you want to find the
defining equation for the plane P through these points. We will as sume that the three points do not all
lie on one line , for otherwise there are many planes through A, B and C.

To find the defining equation we need one point on the plane (we have three of them), and a normal
vector to the plane. A normal vector can be obtained by computing the cross product of two vectors
parallel to the plane. Since and are both parallel to P, the vector is such a normal
vector.

Thus the defining equation for the plane through three given points A, B and C is

45.2 Find the defining equation of the plane P through the points A(2,−1, 0), B(2, 1,−1) and
C(−1, 1, 1). Find the intersections of P with the three coordinate axes , and find the distance from the
origin to P.

Solution: We have

so that

is a normal to the plane. The defining equation for P is therefore

i.e.

The plane intersects the axis when and hence , i.e. in the point . The
intersections with the other two axes are and .

The distance from any point with position vector to P is given by

so the distance from the origin (whose position vector is to P is

45.3. Volume of a parallelepiped

A parallelepiped is a three dimensional body whose sides are parallelograms. For instance, a cube is an
example of a parallelepiped; a rectangular block (whose faces are rectangles, meeting at right angles)
is also a parallelepiped. Any parallelepiped has 8 vertices (corner points), 12 edges and 6 faces.

Let be a parallelepiped. If we call one of the faces, say ABCD, the base of the parallelepiped,
then the other face EFGH is parallel to the base. The height of the parallelepiped is the distance from
any point in EFGH to the base, e.g. to compute the height of one could compute the distance
from the point E (or F, or G, or H) to the plane through ABCD.

The volume of the parallelepiped is given by the formula

Since the base is a parallelogram we know its area is given by

We also know that is a vector perpendicular to the plane through ABCD, i.e. perpendicular
to the base of the parallelepiped. If we let the angle between the edge AE and the normal be ,
then the height of the parallelepiped is given by

Therefore the triple product of is

i.e.

46. Notation

In the next chapter we will be using vectors, so let’s take a minute to summarize the concepts and
notation we have been using.

Given a point in the plane, or in space you can form its position vector. So associated to a point we
have three different objects : the point, its position vector and its coordinates. here is the notation we
use for these:

OBJECT	NOTATION
Point . . . . . . . . . . . . . . . . .	Upper case letters, A, B, etc.
Position vector . . . . . . .	Lowercase letters with an arrow on top. The position vector of the point A should be , so that letters match across changes from upper to lower case.
Coordinates of a point	The coordinates of the point A are the same as the components of its position vector : we use lower case letters with a subscript to indicate which coordinate we have in mind: .

47. PROBLEMS

Computing and drawing vectors

361. Simplify the following

362. If are as in the previous problem , then
which of the following expressions mean anything?
Compute those expressions that are
well defined.

363. Let be three given vectors, and suppose



(a) Simplify and
).
(b) Find numbers r , s, t such that

(c) Find numbers k, l, m such that


364. Prove the Algebraic Properties (49), (50),
(51), and (52) in section 40.2.

365. (a) Does there exist a number x such that

(b) Make a drawing of all points P whose position
vectors are given by

(c) Do there exist a numbers x and y such that

366. Given points A(2, 1) and B(−1, 4) compute
the vector . Is a position vector?

367. Given: points A(2, 1), B(3, 2), C(4, 4) and
D(5, 2). Is ABCD a parallelogram?

368. Given: points A(0, 2, 1), B(0, 3, 2), C(4, 1, 4)
and D.

(a) If ABCD is a parallelogram, then what are
the coordinates of the point D?

(b) If ABDC is a parallelogram, then what are
the coordinates of the point D?

369. You are given three points in the plane:
A has coordinates (2, 3), B has coordinates
(−1, 2) and C has coordinates (4,−1).

(a) Compute the vectors
and

(b) Find the points P, Q, R and S whose position
vectors are ,and , respectively.
Make a precise drawing in figure 21.

370. Have a look at figure 22
(a) Draw the vectors, and


(b) Find real numbers s, t such that
.
(c) Find real numbers p, q such that
.
(d) Find real numbers k, l, m, n such that =
, and  .