DOT and CROSS PRODUCTS Vectors, whether in space or space, can be added, subtracted, scaled, and multiplied. There are two different products, one producing a scalar, the other a vector. Both, however, have important applications to physics and engineering as well as to mathematics. They incorporate geometry, area and volume into vectors as we shall see. The angle between vectors and is the angle shown in the figure to the right by first arranging the vectors so that they have the same tail. Notice that there are really two choices of, one smaller than, the other larger than (unless both equal ). By convention the smaller one is always chosen, so that. It's important to note that the angle between any two of the unit coordinate vectors,, and is because they are mutually perpendicular. Definition: the Dot Product,, quite frequently called the Inner Product, of vectors and is the scalar value defined by where is the angle between and. Since, vectors are perpendicular when and. A number of properties follow immediately from this definition using the perpendicular vectors : Properties: 1. 2. 3. These properties provide a convenient algebraic way of computing the dot product of vectors For by expanding using also Properties 1, 2, and 3, we get
Example 1: determine the dot product of the vectors is given by Solution: the dot product,, of vectors Thus Example 2: determine the dot product of the vectors when coordinate free form by and the angle between is. where is the angle between. When, therefore, Solution: the dot product is defined in PROJECTIONS, COMPONENTS: the geometric definition of dot product helps us express the projection of one vector onto another as well as the component of one vector in the direction of another. But let's approach the concept from a different direction: given vectors and scalars, we know how to form the linear combination to create a new vector. Suppose instead that we start with vectors, and a vector. Then we can try to determine scalars so that In mathematical terms, this provides a representation of in terms of. It is an extremely important idea that occurs everywhere one tries to model a theoretical or practical situation you did it already with Taylor series, for instance, taking for the monomials and a function. The term can be thought of as the projection of on. But for simplicity, let's start with just two vectors and shown below in dark blue and light blue respectively. On the other hand, when we denote by the unit vector in the direction of, then by right triangle trig in the graphic to the right the dashed red vector has assuming, and it points in the direction of. Thus the This red vector is called the projection of on, written
and the scalar is called the component of in the direction of. Example 3: the box shown in Solution: as the unit cube has side length, So when and the projection of on is the vector is the unit cube having one corner at the origin and the coordinate planes for three of its faces. Determine the projection of onto. You'll meet these ideas again in wave motion when you learn about Fourier coefficients and Fourier series, for instance; trig functions, interpreted as harmonics, then play the role of the unit coordinate vectors. In fact, a Taylor Series can be thought of as an infinite dimensional version where the coefficient is the 'component' of a 'vector' in the direction of the basis function. Important Special cases: since the vectors and all have unit length, a vector can be written as On the hand, components of force vectors like gravity or velocity will be important in this and many other courses. CROSS PRODUCT: now we want 'multiplication' of vectors to produce a vector,, not a scalar. Such a vector product occurs many times in geometry as well as in engineering and physics. Recall that the determinant of a matrix is defined by and that of a matrix by the entries being labeled simply to emphasize how they get combined and multiplied.
Because a vector has direction, a convention has to be adopted when defining the 'vector' product of two vectors. If, are vectors arranged so that they have the same tail, then vectors are said to form a right handed system when is perpendicular to the plane containing, and points in the direction shown to the right. Since there could be two directions for to point and still be perpendicular to the plane containing and, the right hand convention amounts to specifying which direction we'll choose. Notice that is a right handed system. More generally, we ask: Question: for a right handed system, (i) is a right handed system? (ii) is a right handed system? Answer: switching (i) NO, (ii) YES, reverses the direction of, so is not right handed is right handed. Definition: the Cross Product,, of vectors and is the vector defined by where is the angle between and, and is the unit vector such that forms a right handed system. Since, vectors are parallel when and.
A number of properties follow immediately from this definition and the fact that is a right handed system: Properties: 1. 2. 3. of COMPUTING CROSS PRODUCTS: the previous properties provide a good algebraic way of computing the cross product For by Properties 1, 2, and 3, so by properties of determinants, Example 4: determine all unit vectors orthogonal to Now Solution: the non zero vectors orthogonal to all of the form are with a scalar. This means the only unit vectors orthogonal to are So Thus the unit vectors orthogonal to are
SCALAR TRIPLE PRODUCT: the dot and cross products of vectors can be combined and written as a determinant: called the Scalar triple product. To illustrate how the cross and scalar triple product get used note: Geometric Application 1: when vectors, are adjacent sides of the parallelogram shown to right, then the height of the parallelogram is, so its Thus the length of are and. is the area of the parallelogram whose sides Geometric Application 2: the vectors, and shown respectively in blue, red and green to the right form adjacent edges of a parallelepiped. But by Geometric Application 1, the base has area, while its Thus the parallelepiped has Mechanics Application 1: when we push down on a bike pedal we exert a force, a vector, on the bike pedal. The objective is to turn the chain wheel. In the figure to the right only the component of will have any effect. Thus Torque as defined by the cross product is a measure of the turning force exerted as the cyclist pushes on the pedal. We shall meet many such applications on a number of occasions. For instance, two useful properties of the scalar triple product follow immediately from its interpretation as the volume of a parallelepiped: because in both cases the parallelepiped collapses to a parallelogram, and so has zero volume. But what might we say about the triple vector product Let's leave this as the first of a set of
CHALLENGE PROBLEMS: when are arbitrary vectors in, Problem 1: establish the identity Problem 2: establish and interpret the identity For the triangle shown to the right: Problem 3: establish and interpret the identity Problem 4: describe in vector terms the foot of the perpendicular from onto side.