Models for showing, from left to right, 1-D, 2-D, 3-D position vectors, circulation, and vectors into and out of the chalkboard.
The block holding the three straight vectors has additional holes along the rim at 45° to either side of vertical, and another opposite the one holding the vector pointing along the rear of the table.

Some physical properties, such as those illustrated in demonstration 04.06 – Meter stick, 1-kg mass, timer, which are length, mass and time, can be specified completely by a number (with appropriate units, of course). These are called scalar properties. The quantities themselves are referred to as scalars, and have magnitude only, for example, 2 meters, 5 kilograms or 10 seconds. Some physical quantities, however, require more than just a single number to represent them. One example is the displacement of an object, that is, its motion from one point to another, for which we need to specify not only the distance between the two points, but the direction in which the object must travel to go from the first point to the second. Such a quantity – one that has both magnitude and direction – is called a vector. Schematically, we represent a vector as an arrow whose length is proportional to its magnitude. In the example of a displacement, if a particle travels from point A to point B, we represent the particle’s travel by a line joining the two points, with an arrowhead at point B. Its magnitude is the distance between the two points, and its direction is that in which point B lies relative to point A along a straight line. It does not matter what path the particle took to get from A to B. As long as it started at A and ended at B, its overall displacement is represented by the vector that connects the two points. Notations vary, but in print, a vector is usually denoted by boldface, for example, a, and its magnitude, a scalar, in regular italic, a. In writing, and sometimes in print, vectors are denoted by a horizontal arrow above the letter, or a “hat” (caret): â. Some examples of vector quantities are velocity, acceleration, force, electric field and magnetic field.

You can add vectors in two (equivalent) ways. The first, sometimes called “head-to-tail addition,” is to draw the first vector, say, a, then draw the second vector, say, b, so that its tail is at the head of vector a. The sum is then the vector that you obtain by drawing an arrow from the tail of a to the head of b, with its tail at the tail of a and its head at the head of b. This vector is called the resultant, which we can denote r. The equation for this is thus a + b = r. By drawing various sets of vectors and adding them, you can show that vectors obey both the commutative law, a + b = b + a, and the associative law, a + (b + c) = (a + b) + c. To subtract, we merely define -b as the vector that has the same magnitude as, but opposite direction to that of b. Thus, a - b = a + (-b). The second, more versatile way, is to choose a coordinate system, resolve the vectors into components, and manipulate those by what is called the analytic or analytical method.

We may choose Cartesian coordinates, and for now consider vectors in only two dimensions. If we extend perpendicular lines to each coordinate axis from the head and tail of a, then subtract (that is, x_head - x_tail, and y_head - y_tail; we need not do this if the tail is at the origin, where x = y = 0), we find the two components, a_x and a_y. We can see that the vector and its components can be arranged as a right triangle, and thus, if θ is the angle at which a sits relative to the (positive) x-axis, its components are a_x = a cos θ and a_y = a sin θ. Depending on the value of θ, these may be positive or negative. The vector’s magnitude is a = √(a_x² + a_y²), and we can find the direction by tan θ = a_y/a_x. To add the vectors a and b, then, we find their components, a_x and a_y, and b_x and b_y, and add: r_x = a_x + b_x, and r_y = a_y + b_y. Then the magnitude is r = √(r_x² + r_y²), and we find the direction from tan θ = r_y/r_x. We can fully specify r either by its components, (r_x, r_y), or by its magnitude and direction, r, tan^-1 (r_y/r_x).

Besides expressing a vector either in terms of its components or as a magnitude and a direction, we can use something called a unit vector. This is a vector that has unit length, and, for this example, points in the direction of our vector r. We can then express r as r = u_rr, where r is the magnitude of r. Similarly, we can use unit vectors pointing in the (positive) directions of the coordinate axes to express the components of a vector. Typically, we use i, j, and k to represent unit vectors pointing in the directions of the x-, y- and z-axes, respectively. The vectors above would thus be a = a_xi + a_yj and b = b_xi + b_yj.

We noted above that we can add vectors, and the rules that such addition obeys. Sometimes, it is useful to multiply a vector, either by a constant, or by a second vector representing a different physical quantity from that of the first, the dimensions of the multiplication giving rise to those corresponding to a third physical quantity. Here we will consider three types of multiplication of a vector.

1) Multiplication by a scalar: We take a vector, a, and multiply by a scalar, k, to get ka. If k is positive, this gives a new vector whose direction is the same as that of a, and whose magnitude is k times that of a. If k is negative, the magnitude is the same (k times the original magnitude of a), but the direction is opposite to the original direction of a. To divide by a scalar, k, we multiply by the reciprocal, 1/k.

2) The scalar product (or dot product) with another vector: The dot product of two vectors, a and b, is written a · b, and defined as ab cos φ, where a and b are the magnitudes of the two vectors, and φ is the (smaller) of the two angles between them (placed with their tails together). We take the smaller angle by convention. For the dot product, though, since cos (2π - φ) = cos φ, we should get the same result for either angle. Since a and b are scalars, and cos φ is a pure number, the dot product is a scalar. We can think of this as the magnitude of one vector multiplied by the component of the other vector in its direction. That is, a cos φ is the projection of a on b, which gives the component of a along b, and b cos φ is the projection of b on a, which gives the component of b along a.

We can also calculate the dot product by multiplying together the components of the two vectors. To do this, we must first realize that i · i = j · j = k · k = (1)(1) cos 0 = 1, and that i · j = j · k = i· k = (1)(1) cos (π/2) = 0. Thus, for a = a_xi + a_yj + a_zk and b = b_xi + b_yj + b_zk, we have a · b = (a_xi + a_yj + a_zk) · (b_xi + b_yj + b_zk). This gives us nine terms, of which six are cross terms, which vanish, leaving a · b = a_xb_xi · i + a_yb_yk · k + a_zb_zk · k, which gives a · b = a_xb_x + a_yb_y + a_zb_z.

An example of a physical quantity that arises from a dot product is mechanical work (= F · d).

3) The vector product (or cross product) with another vector: The cross product of two vectors, a and b, is written a × b, and equals ab sin φ, where a and b are the magnitudes of the two vectors, and φ is the (smaller) of the two angles between them (again, placed with their tails together). As noted above, we take the smaller angle by convention. For the cross product, though, since sin (2π - φ) = -sin φ, choosing the larger angle would switch the sign, and thus the direction, of the resulting vector. The direction of this vector is defined to be perpendicular to the plane in which a and b lie. We find the sense of the vector by means of the right-hand rule. We curl the fingers of our right hand in the direction going from a to b through the angle separating them, keeping the thumb erect. When we do this, our thumb points in the direction of the vector that is the cross product of a and b. This is also like imagining turning a screw having a right-handed thread through the tails of a and b, turning in the direction from a to b. The direction in which the screw advances is the direction of the cross product. As we can see from the note above, reversing the order of the two vectors changes the sign of the result, and a × b = -b × a.

As for the dot product, we can also calculate the cross product by multiplying the components of the two vectors. In this case, however, i × i = j × j = k × k = (1)(1) sin 0 = 0, so the three terms that contain these cross products vanish, leaving the six cross terms, which contain the following cross products: i × j = k, j × i = -k, i × k = -j, k × i = j, j × k = i and k × j = -i. If we multiply all the terms and collect, we end up with: a × b = (a_yb_z - a_zb_y)i + (a_zb_x - a_xb_z)j + (a_xb_y - a_yb_x)k.

Two examples of physical quantities that arise from cross products are torque (τ = r × F) and the force on a charged particle as it moves through a magnetic field (F = qv × B).

References:

1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 15-23.
2) Young, Hugh D. University Physics, 8th Edition (New York: Addison-Wesley Publishing Company, Inc., 1992), pp. 21-22.