Vector Algebra
A vector is formally, an element of a vector space. A vector has two components: a direction (or vector) component and a scalar component. For example, gravity is vector: the direction of gravitational force on the earth is towards the center of the earth and the scalar component is approximately 9.8 meters per second squared. As a rule of thumb, any thing that has both a quantity and a direction component can be represented with a vector. For example, velocity, acceleration, force, and momentum are all vectors. Speed and mass, on the other hand, are quantities without directions. They are not vectors but they are scalars. A scalar is a quantity without a direction. A scalar and a vector can multiply to produce a vector in the same direction as the original vector. For example, if a car heads west at 60 mph then a bus moving at half the car's velocity is moving west at 30 mph. In this case, the scalar (1/2) was multiplied by the vector (60 mph west) and the result is the vector (30 mph west).
To illustrate properties of vectors, let us consider the vector space R3 consisting of all ordered triplets of real numbers. In R3, there is a dot product (also called scalar product) of vectors denoted with the symbol ·. If v = (v1, v2, v3) and w=(w1, w2, w3) are vectors in R3. Then v·w = v1w1 + v2w2 + v3w3. The length of a vector v is defined to be the square root of v·v. In R3, the length of v is denoted by ||v|| and is equal to the distance between (0,0,0) and the point (v1,v2,v3). So the definition of length makes sense. The angle between nonzero vectors v and w is defined to be the unique number a such that the cosine of a equals v·w divided by the length of v times the length of w. This angle happens to equal the angle between the line that contains (0,0,0) and (v1,v2,v3) with the line that contains (0,0,0) and (w1,w2,w3). So this definition also makes sense. A scalar in R3 is just a real number. A vector in R3 can be multiplied by a scalar like so: if r is real number then rv = (rv1, rv2, rv3). The length of rv equals the r times the length of v. Also, (rv)·w = r(v·w). So, the angle between rv and w is the same as the angle between v and w. The vector rv is said to be a scalar multiple of v. Oftentimes, it is convenient to work with vectors that have length one. These are called unit vectors. The set of unit vectors in R3 corresponds naturally to the sphere S2, that it is the set of all points distance one away from the origin. In this way, the sphere is said to be the "set of directions" in R3. If v·w = 0 then v and w are said to be orthogonal. If v and w are also unit vectors, then they are orthonormal.
Vectors can be added together by adding their components. So v + w = (v1 + w1, v2 + w2, v3 + w3). The vector v + w can also be determined by the parallelogram rule. Draw the line segment from (0,0,0) to (v1, v2, v3) and the line segment from (0,0,0) to (w1, w2, w3) in three-dimensional space. Now draw the parallelogram that these two segments determine. The diagonal of the parallelogram, that is the segment from (0,0,0) to (v1 + w1, v2 + w2, v3 + w3) represents the vector v + w. As an application, consider a billiard ball that is moving north at 3 mph. Another billiard ball moving west at 4 mph strikes it and stops. If there is very little friction and air resistance, the first billiard ball will now be moving northwest at about 5 mph. In vector notation, the first billiard's original velocity is given by (0,3), the second's by (-4,0) and the first billiard's final velocity by (-4, 3).
The cross product of two vectors in R3 is written with an X sign. It is defined by v X w = (v2w3 - v3w2, v3w1 - v1w3, v1w2 - v2w1. In other words, if v X w = z = (z1, z2, z3), then the determinant of the matrix whose rows are (a,b,c), v, w is z1a + z2b + z3c. In yet other words, if v and w are linearly independent then v x w is a vector perpendicular to plane spanned by v1 and v2. Its length is equal to the area of the parallelogram spanned by v and w, and its direction is given by the right-hand rule. This rule is the following: open your right hand so that your fingers point in the direction of v. Close your fist in the direction of w and stick your thumb out. Your thumb points in the direction of v X w. Here are the main identities concerning the cross product:
- ||v X w|| = ||v||||w|| sin(a)
- v X w = - w X v
- v X (w + y) = v X w + v X y
- r(v X w) = (rv) X w
- (v X w)·y = the determinant of the matrix whose rows are v, w, and y (in order).
- (v X w) X y = (v·y)w - (w·y)v
There are many ways in which the cross product is used. The definitions of the physical torque and angular momentum, both of which have to do with rotational motion, are defined with the cross product. If a straight wire carrying a current is in a magnetic field then the force exerted by the field on the wire is given by the vector equation F = iL X B in which I is the current, L is the length of the wire and B is the force of the magnetic field. The area of a surface in three-dimensional real space is the integral over the surface of the function (V(x) X W(x))·n(x) with respect to x. Here V and W are orthonormal vector fields and n is the unit normal vector to the surface at x. A vector field is a continuous function from the surface to the vector space R2 and a normal vector is orthogonal to the tangent plane at x.
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