Vectors and Motion Vectors are covered in Chapter 3 of Giancoli, but the treatment of vectors there is somewhat incomplete. To remedy this situation, this handout will provide an overview of vector notation, and some additional topics about vectors not covered in Giancoli. If you ve never seen vectors at all, you should definitely read sections 3-1 through 3-5 of Giancoli before continuing with this handout. Notation Giancoli uses boldface letters with an arrowhead to denote vectors, like this: v. Boldface is great if you typeset everything, but awfully inconvenient for writing by hand, so we ll just use arrowheads to denote any vector quantity. The key is consistency: literally every symbol with an arrowhead over it refers to a vector, and almost anything without an arrowhead symbol is not a vector. (Why almost? Well, there are two exceptions. The first is that the zero vector is generally written as 0 instead of 0. No big deal there. The second is: by convention, unit vectors, which are special vectors with a magnitude of 1 that point in the directions of the coordinate axes, are represented with little hats instead of arrowheads: î = unit vector pointing in x-direction ĵ = unit vector pointing in y-direction ˆk = unit vector pointing in z-direction You can think of the hats as being a special version of the arrowheads.) But there are many scalar quantities that are related to vectors, and these generally are not written with arrowheads: 1. Magnitude. If v is a vector, we ll often want to talk about the magnitude of v, which is a scalar. The clearest and most explicit way of doing so is to use vertical bars for magnitude, just like absolute value: v = some vector v = magnitude of v (a scalar) However, it is also very common to denote the magnitude of v simply with the letter v (with no arrowhead). You should be comfortable with this notation; in particular, note that v is a scalar (no arrowhead), but it is a scalar that is closely related to the vector v. Also, remember that v is always a positive number (or zero); the magnitude of a vector cannot be negative. 2. Components. Given a vector and a coordinate system, we can decompose the vector into its components along the coordinate axes, as described in Section 3-4 of Giancoli. For example, suppose vector a has a magnitude of 10 and points 30 above the positive x-axis, as in Figure 1 at right. Then we can consider the horizontal and vertical parts of a separately. The horizontal component is labeled a x and the
vertical component is denoted a y. If this were a three-dimensional problem, we could also have a z-component a z. The notation is important: we use the letter which names the vector, and indicate which component by using an x, y, or z subscript. Using trig, we can determine the numerical values of the components: a x = 10cos 30 = 5 3 8.7 a y = 10sin 30 = 5 Going the other way (from components to magnitude and direction) isn t hard either: a = a x 2 + a y 2 = 5 3 ( ) 2 + 5 ( ) 2 = 10 = arctan a y a x = arctan 1 3 = 30 There are some very important things to keep in mind when dealing with components: a. Components are scalars. Note that even though a is a vector (and has an arrowhead symbol), a x does not have an arrowhead symbol. It s just a scalar (an ordinary number). If you want to write a vector equation that expresses a in terms of its components, you cannot do the following: a =? a x + a y This equation makes no sense because the left side is a vector, but the right side is the sum of two scalars and is therefore also a scalar. Instead, you have to multiply each component by the appropriate unit vector: a = a x î + a y ˆ That works, because both sides of the equation are vectors. Giancoli is particularly confusing about the distinction between vectors and components, because it introduces the term vector component, which is denoted with both an arrowhead and a subscript, like this: v x. However, in this course, we will never use this term or this notation, and any time we refer to something as being a component, that something will be a scalar. If we really need to refer to the vector which points along x and has magnitude a x, we ll just write it out explicitly as a x î. b. Don t confuse a component with the magnitude. Components can be positive, negative, or zero. This is in contrast with the magnitude of a vector, which cannot be negative. In the example in Figure 1, both a x and a y were positive, but consider what would happen if the vector a pointed 30 below the x-axis instead, as in Figure 2 at right. Then a x is still positive, but a y is now negative (because a points down, and the +y-direction is defined to be up). Similarly, a vector can have a negative x-component if it points to the left (assuming the usual coordinate system where +x is defined to be to the right).
This means that you should use some caution when using the magnitude and direction of a vector to calculate its components. One approach is to always define a vector s direction in terms of the angle θ made between the vector going upward from the positive x-axis; using this convention, the vector a in Figure 1 would have = 30, and the vector in Figure 2 would have = "30 (or equivalently, 330 ). Then you could always apply the equations v x = vcos v y = vsin to get the components. (Remember, v in those equations is the magnitude of v, which can t be negative; but the sine or cosine could be negative.) The drawback of this approach is that you have to remember how to take the sine or cosine of angles larger than 90 or less than 0. It s easier and less mistake-prone to just consider the acute positive angle that any vector makes with the x-axis (even the negative x-axis, for a vector which points to the left), use the above equations, but then manually check the sign afterwards so that vectors which point down get a negative y-component, and vectors which point to the left get a negative x-component. The surest way to tell if a symbol represents a component or the magnitude is: does it have a subscript of x, y, or z? If so, it s a component. Note that subscripts might also be used to distinguish two quantities which share the same letter; for example, we could talk about velocities v 1 and v 2, or about forces F N and F g. Those subscripts don t specify components, so v 1 or F N (with no arrowhead) would be considered a magnitude, not a component. To specify components, we d write v 1x or F Nx. There s one little exception to this notation: if we are working with specifically a position vector r, then instead of writing its x-component as r x, we can just write x. You can think of this as a kind of shorthand. The same goes for y instead of r y and z instead of r z. If the position vector itself is subscripted to distinguish it from other position vectors, like r 1, r 2, etc., then we could talk about the x-component of r 1 as x 1 instead of writing r 1x. Similarly, if we have a displacement vector r, the x-component can be written as just Δx instead of r x. Note that x by itself, even though it has no subscript, is not a magnitude, because there is no vector called x. Rather, it s the x-component of the position vector r, and as such it can be positive or negative. Now, if you don t feel like using the shorthand, feel free not to; there s nothing wrong with writing r x. But you will see the shorthand notation in lectures and the book, so it s important to understand what it means. 3. Components uniquely identify a vector. You ve been told that vectors are defined as a magnitude and a direction, which is true. However, if there is a coordinate system, then any vector can also be specified by giving all of its components. In other words, the three statements below are totally equivalent to each other: a has a magnitude of 10 and points 30 above the positive x-axis a x = 5 3 and a y = 5 a = ( 5 3)î + ( 5) ĵ
You can also specify the components by expressing them as an ordered pair: ( a x,a y ). (In three dimensions, it would be an ordered triple.) So a fourth equivalent statement would be: a = ( 5 3,5) So when a problem asks you to find (or calculate, or specify) a vector quantity, you can either give the magnitude and direction, or you can specify all the components in some coordinate system. Vector Addition Adding vectors is certainly covered in Giancoli, but what the text doesn t make clear is this: whenever you add two (or more) vectors, you should always, always, always resolve the vectors into components and then add the components. Yes, it is straightforward to demonstrate vector addition graphically, but that doesn t actually help you calculate the sum of the vectors. Now, it is possible to calculate the sum of two vectors without using components, but it is difficult; it involves repeated application of the law of sines and the law of cosines (which you probably don t remember and certainly don t need), and nobody ever does it that way. And if you ever have to add three (or more) vectors, it gets ridiculously complicated unless you use components. The moral of the story is: always work with components. If you re given magnitude and direction to start with, resolve into components first, then work the rest of the problem. If you need to find the magnitude and/or direction at the end, do the problem using components, and then convert to magnitude and direction. Vector Multiplication There are actually three different operations that could be called vector multiplication : 1. Multiplication of a vector by a scalar 2. Dot product of two vectors 3. Cross product of two vectors Only the first of these is covered in Giancoli Chapter 3, in section 3-3, and the treatment is pretty brief. (Dot products are covered in section 7-2, and cross products in section 11-2.) So let s discuss them all, even though you won t need to use dot products or cross products until later in the course. 1. Multiplication of a vector by a scalar. Consider a vector v and a scalar c. When you multiply v by c, the resulting product c v has a magnitude equal to c v (that is, the absolute value of c times the magnitude of v ). The direction is the same as the direction of v if c is positive, and the direction opposite to that of v if c is negative. (If c = 0, then c v is the vector 0, which has no direction.)
If you are working with components, it s even simpler: just multiply each component by c. In vector notation, this could be written: c( v x,v y,v z ) = ( cv x,cv y,cv z ). For the most part, you can work with multiplication of a vector by a scalar the same way you do with ordinary multiplication of two scalars. 2. Dot product of two vectors. This is one of the two ways to multiply two vectors together, and we ll see its use in physics when we talk about work and energy. The dot product of two vectors a and b, written as a b, is defined to be a scalar (NB) equal to the magnitude of a, times the magnitude of b, times the cosine of the angle between a and b. In symbols, if a and b are vectors and θ is the angle between them (as shown in Figure 3 at right), then a b = a b cos" = abcos". (It doesn t matter if θ is measured from a to b or from b to a.) Note that a and b are magnitudes, so they are always positive. So the sign of a b is determined by the angle θ, because cosine can be positive, negative, or zero: A. If a and b point generally in the same direction (as in Figure 3), then θ is less than 90, so a b is positive. As a special case of this, if a and b are parallel (i.e. they point in the exact same direction, so that θ = 0), then a b = ab. B. If a and b point generally opposite to each other, then θ is greater than 90, so a b is negative. As a special case of this, if a and b are anti-parallel (i.e. they point in the exact opposite direction, so that θ = 180 ), then a b = "ab. C. If a and b are perpendicular to each other, then θ = 90, so a b is zero. Now, you might remember that we advised you to always use components when working with vectors, and here we ve gone and defined the dot product in terms of magnitudes and directions. Does that mean that when you want to take a dot product, you have to convert from components back to magnitude and direction? Thankfully, no. If a is a vector with components a x,a y,a z ( ), then and b is a vector with components b x,b y,b z a b = a x b x + a y b y + a z b z. ( ) Amazingly, this extremely useful identity holds true no matter which coordinate system you use to express the components, just as long as you use the same coordinate system for both a and b. It s a very handy identity. A word of warning: many students misremember this identity as a b =? ( a x b x,a y b y,a z b z ). Don t make this mistake That equation can t possibly be true, because the right-hand side is a vector (expressed in component form) and the left-hand side is a scalar.
The component identity is often simpler than it looks, because it will almost always be true that one of the two vectors in the dot product will lie directly along a coordinate axis. Why is this the case? For the simple reason that you can choose whatever coordinate system you want, so you generally pick one to make your life as easy as possible. So for example, if you want to take the dot product F L, and L happens to be a vector which points along the +x-direction, then the components of L are just (L, 0, 0). So the dot product is F L = ( F x,f y,f z ) ( L,0,0) = F x L. (Remember, the symbol L by itself with no subscript and no arrowhead means the magnitude of the vector L.) 3. Cross product of two vectors. We won t have to use this until next semester, but there is another way to multiply two vectors: the cross product. The cross product of two vectors a and b, written as a b, is defined to be a vector perpendicular to both a and b, and with a magnitude equal to the magnitude of a, times the magnitude of b, times the sine of the angle between a and b. That s a mouthful, so let s break it up into two parts. First, the magnitude: if θ is the angle between the two vectors, then a b = a b sin" = absin". Note that unlike the dot product, this is zero when the two vectors are parallel (or anti-parallel), and a maximum when they are perpendicular to each other. As for the direction, for any two vectors a and b, assuming they are neither parallel nor anti-parallel, there are actually two directions that a third vector could have and be perpendicular to both a and b. By convention, the direction of the cross product is given by the right-hand rule, which goes like this: if you point the fingers of your right hand in the direction of a, and curl them towards b, your thumb will point in the direction of a b, as shown in Figure 4. Warning 1: You should curl your fingers in the shorter way from a towards b (i.e. the way that involves turning them less than 180 ). If you curl them the long way around, your thumb will point in the opposite direction. Warning 2: If you try it with your left hand, your thumb will point in the opposite direction. This seems like a silly mistake to make, but you wouldn t believe how many times we ve seen righthanded students working on an exam and curling the fingers of their left hand in the air because the right hand has a pencil in it. Put the pencil down and use the right hand The right-hand rule has a strange consequence: a b is not equal to b a The two vectors both have the same magnitude, because absin is equal to basin. But if you play around with the right-hand rule, you can convince yourself that curling from a towards b results in your
thumb pointing one way, but curling them from b towards a results in your thumb pointing in the exact opposite direction. As a result, b a = " a b ( ). So the order of the vectors in a cross product is crucial. Because of the requirement that a b be perpendicular to both a and b, the cross product is inherently three-dimensional, so it can be tough to draw clear diagrams. Furthermore, calculating a general cross-product using components is too messy to even bother with. However, it s definitely useful to know the cross products of the unit vectors. By convention, coordinate axes used in physics are right-handed, meaning that î ˆ = ˆk instead of ˆk. A more complete list: î ˆ = ˆk ˆ ˆk = î ˆk î = ˆ ˆ î = " ˆk ˆk ˆ = "î î ˆk = " ˆ One bit of notation that can make 3-d diagrams manageable: the symbol is used to indicate a vector that points into the page, and the symbol is used to indicate a vector pointing out of the page. So in a right-handed coordinate system, if +x is to the right and +y is up as is usually the case, then +z points out of the page (in the direction of î ˆ, as given by the right-hand rule). In fact, if you have any two vectors a and b that both lie in the xy-plane, their cross product will point along the z-axis. However, it can point either towards +z or z, depending on which way you have to turn to get from a to b. If you can get from a to b by rotating a counterclockwise (by an angle between 0 and 180 ) until it is parallel to b, then a b will point towards +z ( out of the page, if the page is the xy-plane). If you have to rotate a clockwise (or equivalently, rotate a more than 180 ) to get to b, then a b will point towards z ( into the page ). See Figure 5 below for some examples.