1
Vectors

Some physical quantities are described by scalars, e.g., density, temperature, kinetic energy. These are pure numbers, although they do have dimensions. It would make no physical sense to add a density, with dimensions of mass divided by length cubed, to kinetic energy, with dimensions of mass times length squared divided by time squared.

Vectors are mathematical objects that are associated with both a magnitude, described by a number, and a direction. An important property of vectors is that they can be used to represent physical entities such as force, momentum, and displacement. Consequently, the meaning of the vector is (in a sense we will make more precise) independent of how it is represented. For example, if someone punches you in the nose, this is a physical action that could be described by a force vector. The physical action and its result (a sore nose) are independent of the particular coordinate system we use to represent the force vector. Hence, the meaning of the vector is not tied to any particular coordinate system or description. For this reason, we will introduce vectors in coordinate-free form and defer description in terms of particular coordinate systems.

A vector u can be represented as a directed line segment, as shown in Figure 1.1. The length of the vector is its magnitude, and denoted by u or by |u|. Multiplying a vector by a positive scalar a changes the length of the vector but not its orientation. If a > 1, the vector au is longer than u; if a < 1, au is shorter than u. If a is negative, the orientation of the vector is reversed. It is always possible to form a vector of unit magnitude by choosing a = u- 1.

Figure 1.1 Multiplication of a vector by a scalar.

The addition of two vectors u and v can be written as

Although the same symbol is used as for ordinary addition, the meaning here is different. Vectors add according to the parallelogram law shown in Figure 1.2. If the "tails" of the vectors (the ends without arrows) are placed at a point, the sum is the diagonal of the parallelogram with sides formed by the vectors. Alternatively the vectors can be added by placing the "tail" of one at the "head" of the other. The sum is then the vector directed from the free "tail" to the free "head." Implicit in both of these operations is the idea that we are dealing with "free" vectors. In order to add two vectors, they can be moved, keeping the length and orientation, so that the vectors can be connected head to tail. It is clear from the construction in Figure 1.2 that vector addition is commutative:

Figure 1.2 Addition of two vectors.

Note the importance of distinguishing vectors from scalars; without the bold face denoting vectors, equation (1.1) would be incorrect: the magnitude of w is not the sum of the magnitudes of u and v.

The parallelogram rule for vector addition follows from the nature of the physical quantities, e.g., velocity and force, that vectors represent. The rule for addition is an essential element of the definition of a vector that can distinguish them from other quantities that have both length and direction. For example, finite rotations about three orthogonal axes can be characterized by length and magnitude. Finite rotation cannot, however, be a vector because addition is not commutative. To see this, take a book with its front cover up and binding to the left. Looking down on the book, rotate it 90° counterclockwise. Now rotate the book 90° about a horizontal axis counterclockwise looking from the right. The binding should be on the bottom. Performing these two rotations in reverse order will orient the binding toward you.

Hoffmann (1975) relates the story of a tribe that thought spears were vectors because they had length and magnitude. To kill a deer to the northeast, they would throw two spears, one to the north and one to the east, depending on the resultant to strike the deer. Not surprisingly, there is no trace of this tribe, which only confirms the adage that "a little knowledge can be a dangerous thing."

The procedure for vector subtraction follows from multiplication by a scalar and addition. To subtract v from u, first multiply v by - 1, then add - v to u:

There are two ways to multiply vectors: the scalar or dot product and the vector or cross product. The scalar product is given by

where ? is the angle between u and v. As indicated by the name, the result of this operation is a scalar. As shown in Figure 1.3, the scalar product is the magnitude of v multiplied by the projection of u onto v, or vice versa. The definition (1.2) combined with rules for vector addition and multiplication of a vector by a scalar yield the relation

where a and ß are scalars and u1 and u2 are vectors.

Figure 1.3 Scalar product.

If ? = p in (1.2) the two vectors are opposite in sense, i.e., their arrows point in opposite directions. If ? = p/2 or - p/2, the scalar product is zero and the two vectors are orthogonal. Although the scalar product is zero neither u nor v is zero. If, however,

for any vector v then u = 0.

The other way to multiply vectors is the vector or cross product. The result is a vector

The magnitude is w = uvsin (?), where ? is again the angle between u and v. As shown in Figure 1.4, the magnitude of the cross product is equal to the area of the parallelogram formed by u and v. As depicted in Figure 1.5, the direction of w is perpendicular to the plane formed by u and v and the sense is given by the right hand rule: If the fingers of the right hand are in the direction of u and then curled in the direction of v, the thumb of the right hand is in the direction of w. The three vectors u, v, and are said to form a right-handed system.

Figure 1.4 Magnitude of the vector or cross product.

Figure 1.5 Direction of vector or cross product.

The triple scalar product (u × v) · w is equal to the volume of the parallelepiped formed by u, v, and w if they are right-handed and the negative of the volume if they are not (Figure 1.6). The parentheses in this expression may be omitted because it makes no sense if the dot product is taken first: the result is a scalar and the cross product is an operation between two vectors.

Figure 1.6 Triple scalar product.

Now consider the triple vector product u ×(v × w). The vector v × w must be perpendicular to the plane containing v and w. Hence, the vector product of v × w with another vector u must result in a vector that is in the plane of v and w. Consequently, the result of this operation may be represented as

where a and ß are scalars.

1.1 Examples

1.1.1

Show that if the triple scalar product vanishes

the three vectors are coplanar.

The scalar product u × v is perpendicular to u and v. If the triple scalar product vanishes, then w is perpendicular to u × v and hence is in the plane of u and v. Consequently, w can be expressed as a linear combination of the other two, e.g., w = au + ßv where a and ß are scalars (as long as u and v are not collinear).

1.1.2

Show that if w = au + ßv the triple scalar product of the three vectors vanishes.

Substituting w into (1.6) yields zero because the scalar products of u × v with v and with u are zero.

Figure 1.7 Diagram for Problem 1.6.

Figure 1.8 Diagram for Problem 1.7.

Figure 1.9 Diagram for Problem 1.10.

Exercises

1. Explain (in words and/or diagrams) why

   and that

   where w = u × v.

2. Explain (in words and/or diagrams) why

   but that a minus sign is introduced if the order of any two vectors is reversed.

3. Explain why u × (v × u) is orthogonal to u and show that a and ß in (1.5) are then related by

   where ? is the angle between u and v.

4. Prove that if (1.3) is satisfied for any vector v then u =...