5-21-98
In this course, we'll be dealing with two kinds of quantities, scalars and vectors. A scalar is something that can be specified as just a number, like temperature ( -30 °C) or mass ( 200 g ). A vector requires both a number and a direction. A velocity is a good example of a vector. If you came to campus on the T today, at some point you may have been travelling 20 km/hr east. Velocity is a combination of a scalar (speed, 20 km/hr) and a direction (east).
Examples of scalars : mass, temperature, speed, distance
Examples of vectors : displacement, velocity, acceleration, force
One crucial difference between scalars and vectors involves the use of plus and minus signs. A scalar with a negative sign means something very different from a scalar with a plus sign; +30 °C feels an awful lot different than -30 °C, for example. With a vector, however, the sign simply tells you about the direction of the vector. If you're travelling with a velocity of 20 km/hr east, it means you're travelling east, and your speed is 20 km/hr. A velocity of -20 km/hr east also means that you're travelling at a speed of 20 km/hr, but in the direction opposite to east...20 km/hr west, in other words. With a vector, the negative sign can always be incorporated into the direction.
Note that a vector will normally be written in bold, like this : A. A scalar, like the magnitude of the vector, will not be in bold face (e.g., A).
A vector pointing in a random direction in the x-y plane has x and y components: it can be split into two vectors, one entirely in the x-direction (the x-component) and one entirely in the y-direction (the y-component). Added together, the two components give the original vector.
The easiest way to add or subtract vectors, which is often required in physics, is to add or subtract vector components. This requires breaking up a vector into its components, which involves nothing more complicated than the trigonometry associated with a right-angled triangle.
Consider the following example. A vector, which we will call A, has a length of 5.00 cm and points at an angle of 25.0° above the negative x-axis, as shown in the diagram. The x and y components of A, Ax and Ay are found by drawing right-angled triangles, as shown. Only one right-angled triangle is actually necessary; the two shown in the diagram are identical.
Knowing the length of A, and the angle of 25.0°, Ax and Ay can be found by re-arranging the expressions for sin and cos.
Note that this analysis, using trigonometry, produces just the magnitudes of the vectors Ax and Ay. The directions can be found by looking on the diagram. Usually, positive x is to the right of the origin; Ax points left, so it is negative:
Ax = -4.53 cm in the x-direction (or, 4.53 cm in the negative x-direction)
Positive y is generally up; Ay is directed up, so it is positive:
Ay = 2.11 cm in the y-direction
The textbook deals with the simple cases of adding vectors that are parallel to each other, or at right angles. In general, however, the angle between vectors being added (or subtracted) will be something other than 0, 90, or 180°. Consider the following example, where the vector C equals A + B.
A has a length of 5.00 cm and points at an angle of 25.0° above the negative x-axis. B has a length of 7.00 cm and points at an angle of 40.0° above the positive x-axis. If C = A + B, what is the magnitude and direction of C? There are basically two ways to do this. One way is to draw a picture; we can get a rough idea of what direction C points and how long it is by drawing a vector diagram, moving the tail of B to the head of A, or vice versa. The vector C will then extend from the origin to wherever the tip of the second vector is.
The second way to find the magnitude and direction of C we'll use a lot in this course, because we'll often have vector equations of the form C =A + B. The simplest way to solve any vector equation is to split it up into one-dimensional equations, one equation for each dimension of the vector. In this case we're working in two dimensions, so the one vector equation can be replaced by two equations:
In the x-direction : Cx = Ax + Bx
In the y-direction : Cy = Ay + By
In other words, to find the magnitude and direction of C, the vectors A and B are split into components. The components are:
Ax = -4.532 cm in the x-direction
Ay = 2.113 cm in the y-direction
Bx = 7.00 cos40 = 5.362 cm
By = 7.00 sin40 = 4.500 cm
Bx = 5.362 cm in the x-direction
By = 4.500 cm in the y-direction
The components of C are found by adding the components of A and B:
Cx = Ax + Bx = (-4.532 + 5.362) cm in the x-direction = 0.83 cm in the x-direction
Cy = Ay + By= (2.113 + 4.500) cm in the y-direction = 6.61 cm in the y-direction
The magnitude of C can be found from its components using the Pythagorean theorem:
The direction of C can be found by taking the inverse tangent of Cy/Cx:
inverse tan of 6.613 / 0.830 = 82.8°.
Combined, this gives C = 6.66 cm at an angle of 82.8° above the positive x-axis.
Note how the calculations and the diagrams go hand-in-hand. This will often be the case; it is always a good idea to draw diagrams as you go along.
Note also that we could have used the cosine law to get the length of C, and then applied the sine law, with a bit of geometry, to get the angle. It's worth trying that for yourself, just to convince yourself that the numbers come out the same.