(c) copyright Foundation Coalition (S. A. Fulling) 1997

Class 6.1

Vectors and the Dot Product

Reading assignment for Monday, October 6

Stewart 11.2 and 11.3
This Web page

We live in a multidimensional world, and first-year calculus has begun to recognize that fact. From the beginning we need to handle physical quantities that have a direction as well as a size, and therefore need to be represented by more than one numerical component. Fortunately, most first-semester physics problems involve in any essential way only two dimensions, not three. Therefore, for the present we shall concentrate on two-dimensional vectors, which are easier to draw and visualize. Most of the elementary facts about vectors are the same in all dimensions (greater than 1), so you should have no trouble reading the three-dimensional discussion in the textbook for today.

Two kinds of vectors

From a physical or geometrical point of view, "a vector is an arrow" with a certain length and a certain direction, and that concept is more fundamental than the representation of the vector by a string of 2 or 3 numbers. Students often have trouble understanding where the tail of the arrow belongs. It needs to be pointed out that in physics two different kinds of vectors are used. (These correspond to the notations "(.,.)" and "<.,.>" in our calculus textbook.)

The position of a particle is represented by a vector with its head at the particle and its tail at some arbitrary fixed point chosen as the origin of a coordinate system. When we draw axes through the origin and mark off units on the axes, the vector becomes represented by a pair of numbers, the coordinates of the point relative to that coordinate system. If the origin of coordinates is changed, the vector and its coordinates change.
[insert drawing] (temporary screen with both pictures)
In this example the vector or point has coordinates (3,2) relative to the original origin, O. If the origin is moved to O', which has coordinates (1,-0.5) with respect to O, then the point has coordinates (2,2.5) with respect to O', and the arrow representing it has changed. Notice that the new coordinate vector is obtained by vector subtraction of the coordinate vector of O' from the old coordinate vector of the point.
Almost all other vectors in physics, such as velocity, acceleration, force, electric field, ... are drawn with their tails at the point or particle concerned. It is as if each particle carries its little private coordinate system along with it, with its origin sitting on the particle. Although the axes coming out of the particle are never drawn explicitly, the numerical components of these vectors must be read off from these invisible axes, not from the axes of the "global" coordinate system used for positions. Furthermore, if the origin of the global coordinates is changed (but its axes are not rotated), then these vectors and their coordinates stay the same!
[insert drawing] (temporary screen with both pictures)
Here we have drawn a velocity or force vector at our example point, with coordinates or components <-1.5,-1> in appropriate units. Both the arrow and the numbers <-1.5,-1> are unchanged if the origin is moved from O to O'.
The numerical parts a_i of a vector < a₁,a₂> are sometimes called coordinates and sometimes components. Most mathematics texts call them "coordinates" and reserve "components" for the corresponding vectorial pieces, a₁i and a₂j, where i and j are the unit vectors in the x and y directions. (Note that < a₁,a₂> = a₁i + a₂j.) However, Stewart and most physics texts call these numbers "components" and reserve "coordinates" for the numbers representing vectors of the position type.

Unit vectors and projections

Given any vector, we can construct a vector of length 1 that points in the same direction, simply by dividing the original vector by its own length. For example, if a = <3,-4>, then its length, |a|, is 5 and the corresponding unit vector is

u = a/|a| = <3/5,-4/5> (or <3,-4>/5).

(A common notation for the unit vector in the direction of a is an a with a caret ("hat") over it; unfortunately, this (like the raised dot in the dot product) is difficult to produce on the Web.)

Now, given another vector, b, we can decompose b into a piece in the direction of a and another piece perpendicular to a. (Please see p. 699 of Stewart for sketches.) The construction of the piece parallel to a takes place in two steps:

The number u.b is called the scalar projection of b onto a (or onto u).
The vector (u.b) u is called the vector projection of b onto a (or onto u).

Both projections are commonly called "the component of b along a".

Stewart writes the formulas for the projections in terms of the original vector a (resulting in factors of |a| and |a|² in the denominators). In the opinion of your current instructors, the formulas in terms of u are better: Not only are they simpler to write, but they are closer to the intuitive meaning of the projection. Only the direction of the vector a is relevant, and that direction is best represented by the unit vector.

Exercise: Find a unit vector v perpendicular to a = <3,-4>. (There are two possible correct answers. Why?) Then show that any vector b is the sum of the vector projections of b onto u and v.

Orthogonal bases

("Orthogonal" is just a fancy name for "perpendicular".)

We remarked earlier that every vector can be decomposed into the sum of its vector components along the axes -- that is, its vector projections onto the basic unit vectors i and j. In the foregoing exercise you saw that the same thing is true of the perpendicular unit vectors u and v. If you draw u and v on a coordinate grid, you will see that they look exactly like i and j except for being rotated. We can think of u and v (or v and u, depending on which overall sign you chose for your v) as the basic unit vectors corresponding to some rotated coordinate system.

Theorem: Let u and v be any two vectors (in 2-dimensional space) satisfying

|u| = 1 = |v| and u.v = 0.

Then any other vector, b, in that 2-dimensional space is the sum of its projections onto u and v:

b = (u.b) u + (v.b) v.

u and v are called an orthonormal basis for the 2-dimensional space, or just "a set of perpendicular unit vectors". It should not be a big surprise that to do this sort of thing in 3-dimensional space, you will need three mutually perpendicular unit vectors. (We'll get to that next semester.)

Exercise: The vectors

a = <2,-4> and b = <4,2>

in Exercise 20 on p. 701 are orthogonal (but not of unit length). Decompose the vector c = <1,3> into its components in the directions of a and b. Check your answer (do the components add up to c?).