PHYS 2211: Vector Notation Rules of the Road

Overview Vectors are everywhere in physics, and common in many other fields of science and engineering. Although superficially an easy idea—a vector is something that “has a direction”—we find that students very often struggle with vector operations. In most cases, the errors we see can be attributed to carelessness—not because students are sloppy or lazy, but because they underestimate the amount of attention to detail that they must routinely bring to vector operations. Successful work with vector quantities requires you to think carefully about what you are writing, and to adopt a habitual notation for vectors that is as close as possible to being foolproof. These notes provide standardized rules for expressing vector relationships throughout the term. You will find that following these rules will require extra work up front, to make these procedures a matter of habit. Rest assured that it will be time well spent; with practice, using careful vector notation adds little in terms of time required, but adds a lot in terms of clarifying what you are doing.

Rule 1: Coordinate Systems — Before beginning any work on a problem, you should explicitly indicate the coordinate system that you will be using for the problem. It suffices to simply make a brief verbal description, or display an explicit sketch of your chosen axes.

You cannot describe motion without first specifying what reference frame you are measuring against. Don’t just “assume” a coordinate system—you cannot guarantee that those reading your work will make the same assumption. Don’t give anyone the chance to misinterpret your work. Even in 1D situations, vector directions are assigned on a positive/negative basis; such assignments are ambiguous unless you have previously indicated what you mean by “positively-directed”. On tests, you may see points deducted on vector problems, if you do not not begin with a clear definition of the coordinate system that will be used in the problem. Remarks: We routinely see students make significant vector mistakes—because they erroneously think that they will be okay just assuming “the obvious” coordinate system, and don’t need to waste time by initially specifying a coordinate choice. We are confident that if you make an overt choice at the outset of a problem, you will be much less likely to make subsequent oversight errors.

Rule 2: Vectors are Vectors, Scalars are Scalars — The notation used to denote a vector quantity should always look different, on the page, from the notation for a scalar quantity.

Vector math operations can look deceivingly similar to ordinary math; be aware that there are some subtle—but critical—extra considerations when working with vectors. These considerations are easy to deal with if one thinks about them, but are just as easy to overlook if one is incautious. The intent of this rule is to enforce a habit of writing vectors a little bit differently than scalars, so that you will remember to think about them differently, as well. When you are about to write something down, first ask yourself, “is this a vector or scalar relationship?” If it is a vector identity, make it look like one at the outset. The easiest way to accomplish this is to: (a) initially write out general symbolic expressions, without any numerical assignments (we’ll explain how we want you to assign values to a vector in a bit); and (b) make sure that every vector quantity is written as an “arrowed” symbol: e.g., make it a habit to write 𝑣f = 𝑣i + 𝑎 t, rather than just vf = vi + a t. This might seem like a little thing, but it ultimately is a big reminder that there is an extra thinking step that you will need to perform with this equation.

This step is particularly important in 1D situations, which is where vector errors are most common—usually because students don’t remember to keep the vector aspects of the expression in mind. We will do our best in lecture to adopt notation that emphasizes those vector equations requiring extra care. Please do your best to adopt the same considerations into your everyday work. Remarks: The author of the textbook often takes a casual attitude about vector notation in 1D. Keep in mind that his long experience with physics allows him to recognize—without consciously thinking about it—whether a solitary symbol “v” is just a magnitude (making it a scalar), or whether it also implicitly has a sign that conveys a direction (making it a vector). You do not have the experience to reliably intuit the nature of a stand-alone symbol, on sight. Consequently, you should take extra care to make sure that you put the correct directional information into an initial vector statement, and to extract the correct direction information out of a final vector answer. On test questions requiring vector answers, you may see points deducted if the answer you provide is not expressed using an acceptable form of vector notion.

Rule 3: Assigning Vector Values in 1D — In this course, a generic 1D vector should be denoted as an arrowed symbol (e.g. 𝑣); a specific value can then be assigned to a vector by: (a) first specifying a magnitude to denote the size of the vector; then (b) assigning an explicit sign (+ or –) to convey its direction; and finally (c) setting the magnitude and direction together inside angled brackets: 〈 〉 .

For example, suppose we are given a problem that states, “a ball is thrown straight upward with an initial speed of 4.0 m/s.” We would transform this verbal statement into a mathematical expression by performing the following steps: first, establish our coordinate system, in adherence to Rule 1, by stating clearly “let upward be the positive direction”. (We could have also done this non-verbally by simply drawing an explicit coordinate axis pointing upward, on the page.) With this coordinate system, we write:

𝑣! =   +4.0  m/s = +vo

On the left is the generic vector to which we are assigning the value; in this case, we remind ourselves that we are assigning an initial velocity for the ball. In the middle of the statement, we see a numerical magnitude (initial speed) and an explicit positive sign, telling us that the vector is directed in the positive sense along chosen axis. The angled bracket makes the vector nature of that term stand out, in effect stating, “interpret the sign you see here as a statement about direction”. The use of angled bracket for 1D vector assignments is preferred, but other formats are acceptable. On tests, you may see points deducted if there is no clear notation that associates “sign and magnitude” together, as a single vector entity. On the far right of the preceding example, we see that the numerical value for initial speed has been replaced by a symbolic value—which can be advantageous if there will be significant additional calculations to perform. (See the class notes regarding Dimensional Analysis for more detail.) Finally, in considering the example above, we’ll make two more points:

1. Even though we know “𝑣i is up” and “up is positive” we do not leave off the positive sign inside the brackets. In this course, the expressions <v>  and <+v> are not synonymous. We will expect every positively-directed vector to explicitly include the positive sign, as an indicator that you are intentionally assigning that direction. This idea can be summarized as: “Direction: if you KNOW it, SHOW it.”

2. When a numerical value is replaced by a symbol, the symbol itself implicitly carries along both the numerical value and the units. That means, when converting over to a symbolic representation of a vector, there is no need to write “𝑣i = <+vo>  m/s”. Inclusion of the “m/s” is redundant; the symbol “vo” represents a speed, and necessarily carries a physical dimension of [Length / Time], meaning that it already has the necessary units built in.

If you don’t know the direction of a particular vector, the best thing to do is to leave that vector in generic form (i.e. a just symbol with an arrow over it). Such an expression tells the reader, “I know this is a vector, I just don’t have any specific data for it yet”. Just don’t leave off the arrow, because in this course, that would mean, “I claim this is a scalar, with no directional info.”

Rule 4: Adding or Subtracting Vectors — The notation for a sum of vectors should always make it clear that the operation is an addition, and the notation for a difference of vectors should always make it clear that the operation is a subtraction.

This statement might seem superfluous, but consider the following two vector expressions:

+𝑣! + −𝑣!

+𝑣! − +𝑣! Are these expressions equivalent? In physics, the answer is no—they describe different physical operations. The first expression involves computing the sum of two vectors that point in opposite directions, while the second describes computing the difference between two vectors that point in the same direction. The fact that the final math evaluates the same in both cases does not mean that the physics means the same thing. With vectors, addition and subtraction involve more than just “math”—because the math we use is short-hand notation for the operation of combining two vectors to get a third. In certain situations (for example, in Newton’s 2nd Law), our objective is to add two or more vectors, to yield a net, or resultant vector. In other situations (such as computing the change in a vector), we look at the difference between initial and final vectors. Thus, a generic example of vector addition might be:

𝐹 = 𝐹! + 𝐹! + 𝐹! +⋯ = 𝑚𝑎   while a vector subtraction might be:

Δ𝑣 = 𝑣! − 𝑣! In summation, each vector is individually assigned a magnitude and direction, inside angled brackets (as per Rule 3), and then the addition involves an explicit sum of each set of brackets. For example:

𝐹! + 𝐹! + 𝐹! = +𝑓! + −𝑓! + +𝑓!

describes the addition of three forces, having respective magnitudes f1, f2, and f3, in which the first and third vectors are positively directed, and the second is negatively directed. In the case of subtraction, we again start by assigning magnitude and direction for each vector, placing them inside angled brackets, and then explicitly performing a subtraction of the brackets. For example, if an initial velocity has magnitude vo and a final velocity has magnitude v1, then the

vector expression for change in velocity would, depend on the relative directions of the two vectors. If the two vectors are both positively-directed, we would write:

Δ𝑣 = +v1 − +vo On the other hand, if the initial velocity was negatively directed and the final velocity was positively directed, we would write:

Δ𝑣 = +v1 − −vo So, with both addition and subtraction, the steps are: (1) express each vector as a bracket; then (2) perform the specified operation on the entire brackets, not just the individual symbols. Remarks: This requirement might seem like we are being nit-picky about mathematical semantics, but a common source of vector math mistakes involve sign errors—and in most cases, those errors can be traced back to flawed vector assignments at the outset of a problem, when writing out sums or differences. Consequently, using meticulous notation at the beginning of a vector problem can dramatically reduce the chance of vector errors down the line.

Rule 5: Intermediate Algebraic Operations — Provided the initial statement of a vector relationship meets the conditions of Rules 1–4, formal vector notation may be relaxed during the process of actually performing subsequent mathematical operations.

We know that tracking formal notation all the way through a problem would be tedious. Since our main goal is to ensure that you start and end vector problems with proper care, we won’t insist that you write every step of a solution in nit-picky vector format. Once the initial assignments are made (i.e. directions and magnitudes, in brackets), you may drop the bracket notation in the next line without penalty. Make sure to preserve all signs, and note that the symbols you are now working with are all magnitudes—which means that they should all be inherently positive scalars. At the conclusion of your solution, you may need to rewrite your final answer back in vector format, if the problem is asking for a vector answer. Here’s an example of how this rule works in practice: A ball is thrown straight up off the roof on a building with an initial speed vo. The ball is in free-fall, with a downward-directed acceleration of magnitude g (where g  = 9.8 m/s2). After what elapsed time will we find the ball travelling downward with the same speed vo that it was initially rising? Free-fall problems involve constant acceleration, with velocities governed by the vector equation 𝑣! = 𝑣! + 𝑎  Δ𝑡. Using a coordinate system where upward = positive, we write: 𝑣! = +vo and 𝑎 = −𝑔 . Our goal? Find the elapsed time Δt that will give us 𝑣! = −vo (i.e. moving at the same speed, but in the opposite direction). Substituting all this into the equation of motion gives: 𝑣! = 𝑣! + 𝑎  Δ𝑡 (generic vector equation)

−vo = +vo + −𝑔  Δ𝑡 (after vector assignments) Note that we are describing a situation in which the ball actually reverses direction some time during the interval Δt. For part of the time the ball is travelling upward, and for part of the time it is travelling downward—the use of vector expressions allows us to include both parts of the motion (“moving up” and “moving down”) in a single equation. We do not need to break off the “moving up” part of the problem from the “moving down” part (something that inexperienced physics

students feel obligated to do). The vector equation handles the direction reversal automatically, if we are careful to express each vector term properly. Having made our initial vector assignments in bracket form, we can drop the formal notation:

−vo = +vo − 𝑔  Δ𝑡 (no brackets, but all signs preserved)

−2vo = −𝑔  Δ𝑡 (subtracting a “vo” from each side)

Before we finish the math, recall that we arranged it such that the symbols vo and g represent magnitudes—which means they are positive scalars. Why is that such a big deal? Looking at the equation −2vo = −𝑔  Δ𝑡, notice that the only way the signs on both sides can make sense is if the unknown Δt is also positive. This should be no surprise; elapsed time should be positive, here. The point is, we can see whether the math is consistent with what the physics says it should be, before we work all the way through the math. Now, finishing our calculations, we have:

2vo = 𝑔  Δ𝑡 so that

Δ𝑡 =2vo𝑔

Since the problem asked for elapsed time—a scalar—we do not need to rewrite the final answer as a vector. (At this stage, a careful student would also make a quick evaluation using dimensional analysis, to verify that the final expression 2vo/g has units of time [T].)

Rule 6a: Magnitude & Direction in 2D — In 2D situations a pair of coordinate axes will be established—and “direction” can be anywhere in that 2D plane. In such situations, a value is assigned to a vector by: (a) specifying the magnitude of the vector; and (b) identifying an angle less than 90°, measured from one of the nearby coordinate axes, to the vector itself.

Many students have been taught to always measure angles counter-clockwise, starting from the positive x-axis. Instead, we will assign direction angles as follows: a vector in any quadrant will have two nearby axes (for example, in Quadrant II of a standard xy-coordinate system, the positive y-axis and the negative x-axis will be the nearest two axes). Specify the angle that the vector makes with either one of the two nearby axes. That angle will necessarily be less than 90°, and hence the sine and cosine functions will always yield positive values. Regardless of whether you trace out the angle in the clockwise or counter-clockwise direction, always assign a positive angle. For example, you might express a velocity vector that lies in Quadrant II as “3.0 m/s @ 25° above the negative x-axis” or as “3.0 m/s @ 65° ccw from the positive y-axis”—both are describing the exact same vector. Just make sure that you make the reference axis, and the direction from that axis, absolutely clear to the reader; “3.0 m/s @ 25°” is an insufficient description of the vector, without a clear sketch indicating how the angle is being measured. Similarly, in map problems, where directions are based off the primary compass headings, you might specify a direction as “an angle θ south of west”— which means, “starting from the due westward direction, rotate around toward the southward direction by an angle θ, to obtain the direction of the vector”. You will find that if you follow these guidelines for assigning direction angles, the process of vector decomposition (Rule 6b below) will be greatly simplified.

What about specifying a direction in 3D situations, instead of 2D? In that case, two angles would actually be needed—the standard convention being the use of spherical polar coordinate angles (θ,φ). That being said, don’t worry—we won’t be dealing with 3D vectors that way, in this course. Instead, our approach for such vectors will be…

Rule 6b: 2D Vectors in Cartesian Form — A general vector in 2D can be described by decomposing it along the two Cartesian axes that make up the coordinate system. One thus writes 𝑣 = 𝑣! + 𝑣!, where 𝑣! and 𝑣! are 1D vectors along the two coordinate axes, with values assigned as per Rule 3. (A 3D vector would also have a component 𝑣!.)

That is, the vector 𝑣 is visualized as the hypotenuse of a right triangle, with the 1D component vectors 𝑣! and 𝑣! being realized as the sides of the triangle, parallel to the coordinate axes. The component 𝑣! is assigned a value by giving it a magnitude (that is, it’s projected size along the x-axis) and an explicit sign (giving the direction that it lies along the x-axis), grouped together inside an angled bracket—and a similar independent assignment is made for 𝑣!. A key problem, however, is to adopt a notation that tells at a glance whether a given expression is describing a general 2D vector, an x-component only, or a y-component only. Keeping components distinct is ABSOULTELY critical. (Failure to do so is the second-biggest source for student mistakes, after sign errors.) In order to avoid such errors, we ask you to adopt the following guidelines in this course:

• General vectors (i.e. symbols-with-arrows): a vector without subscripts, 𝑣, represents an un-decomposed vector that may point in any direction in space. Component vectors along the Cartesian axes should always appear with subscripts: e.g. 𝑣! and 𝑣!, and possibly 𝑣!.

For example, Newton’s 2nd Law reads 𝐹 = 𝑚𝑎. When written in this form, it involves the full 2D (or 3D) representation of each vector, summed as vectors to yield a 2D (or 3D) acceleration vector. However, a key feature of the 2nd law is that it can be broken down into independent component equations:

𝐹! = 𝑚𝑎! and 𝐹! = 𝑚𝑎!

The x- and y-subscripts identify each equation as just a 1D component of the full 2nd Law—each handled as an independent equation, and each separately using the notation specified in the 1D rules, above. (This will, in fact, be the standard way in which we tackle the 2nd law, in this course.) Similarly, 2D projectile motion is a constant acceleration problem, in general. As we’ve seen, in these problems the velocity satisfies 𝑣f = 𝑣i + 𝑎 t —which, written this way, is an equation involving 2D vectors. It can be broken into two separate component equations:

𝑣x,f = 𝑣x,i + 𝑎! t and 𝑣y,f = 𝑣y,i + 𝑎y t

In projectile motion, the constant acceleration vector is precisely the free-fall acceleration, now expressed in terms of components: "𝑎 is downward, with magnitude g” becomes “ 𝑎! = 0 and 𝑎! = −𝑔 ” (where the positive y-axis is upward). With these substitutions, the component-form velocity equations become:

𝑣!,f = 𝑣!,i + 0 t and 𝑣!,f = 𝑣!,i + −𝑔 t

𝑣!,f = 𝑣!,i and 𝑣!,f = 𝑣!,i + −𝑔 t

So in projectile motion problems, the horizontal velocity component remains constant, while the vertical component behaves just like a 1D free-fall component—but x and y are always kept separate!

• Values are assigned to components of 2D vectors using standard 1D notation—sign and magnitude, in brackets—for each component direction, with the added requirement that there is some indication as to which component (x or y) is being assigned.

Consider this projectile problem: a cannonball is fired with a speed vo at an angle θ above the rightward horizontal. We wish to assign a value to the initial velocity, in component form, so that we can apply the velocity equation on the prior page. We start off writing: 𝑣i = 𝑣!,i + 𝑣!,i, and then apply a little bit of trigonometry to express the components as:

𝑣!,i = +vo  cos𝜃 and 𝑣!,i = +vo  sin𝜃

(This assumes a coordinate system with “to the right” as the positive x-direction and “up” being the positive y-direction.) What happens when we substitute these expressions into the original 2D equation? We get:

𝑣i= +vo cos𝜃 + +vo sin𝜃

Suddenly, there is a great deal of ambiguity on the page…there is nothing to prevent the mathematical operation of combining the two terms on the right into one—but the absolute rule is that different components CANNOT be combined in such a manner! Some notation must be adopted that will differentiate x- and y-components. There are several ways that you may do so:

A. Segregation: Keep different components in separate equations. For example, if you are working a 2nd Law problem, it is standard procedure to decompose all forces, and then write separate component equations. Make sure the first line of any calculations is clear about which particular component direction is being described. For example, for a 2nd Law problem in which there is no acceleration along the x-axis, we might have:

𝐹! = 𝑚𝑎!

+𝐹!"#! + −𝐹!"#$ + +𝐹!"#$ cos𝜃 = 𝑚 0

There is no need in the second line to add any extra notation, because the first line makes it clear that what is written says nothing about the y-direction; the bracketed vectors cannot be misinterpreted.

B. Subscripts: When a chance for confusion exists, you may put a subscript on the bracket.

For example, we could write the component expression for 𝑣i in the cannonball problem as:

𝑣i = +vo cos𝜃 ! + +vo sin𝜃 !

No chance for confusion there—we’ve written something that makes it clear to the reader—and to ourselves—that the two brackets are describing different coordinate directions and cannot be mathematically combined. In fact, keeping x- and y-components separate in a single equation is important enough that a special notation is used in physics: unit vector notation. The symbol 𝜄 (read as “i–hat”) will be used as a shorthand notation to denote the phrase “parallel to the x-direction”, the symbol 𝚥 (“j–hat”) will denote the phrase “parallel to the y-direction”, and the symbol 𝑘 (“k–hat”) will denote “parallel to the z-axis”.

C. Unit Vectors: Component directions may be identified by placing the appropriate unit vector after the bracket that assigns a value to the component.

Thus, we could write a one-line expression for the initial velocity vector for the cannonball as:

𝑣i = +vo cos𝜃  𝜄 + +vo sin𝜃  𝚥

The use of unit vector notation (as above) will be the preferred form for making component vector assignments in this course. The utility of this notation is that it allows a quick, at-a-glance mechanism for sorting together parallel components and sorting apart perpendicular ones. At this point, we need to make a careful distinction. Once we start using unit vectors, the symbols 𝜄 and 𝚥 are doing the actual pointing, so they are viewed as being the real “vector parts” of any expression. The brackets contain sign and magnitude data—but technically we now view that to be “not-vector” information. (That is, it doesn’t have its own direction.) When we write something like 𝑣!,i = +vo  cos𝜃  𝜄, we think of the vector part as being “  +𝜄  ” (a stand-alone vector of length one, parallel to the positive x-axis), while the factor “  vo cos𝜃  ” just gives us a size:

↓ this is the vector part, that conveys direction 𝑣i,! = (vo cos𝜃)    +𝜄    

↑ this is not the vector part, but it is important

Rather than splitting hairs and having to rearrange our notation, we will continue to write vectors in the form: 𝑣!,i = +vo  cos𝜃  𝜄, understanding that 𝜄 is the real vector, and the stuff in brackets just “fills in the details”. The bracket notation is kept, reminding us to be explicit about which way (positive or negative) the vector points along an axis, but the bracket itself is not seen as a vector. In words, our notation means: “ 〈positively-directed, of magnitude vo cosθ 〉, along the x-axis ”. The underlined portion is the actual vector part of the statement. Finally, another notation that is occasionally used for components—most often in computing applications—is an ordered pair expression (or ordered triplet, in 3D). Since 𝜄 and 𝚥 are essentially just place-keepers, we recognize that simply setting a standard order for two components can avoid ambiguities:

D. Ordered Pairs: Different components can be distinguished in a single equation by assigning values as an ordered pair in angled brackets.

Each term inside the brackets must have an explicit sign for direction, as usual for 1D components. Thus, the expression 𝑣i = +vo cos𝜃  ,+vo sin𝜃 would be another valid way to express the initial velocity city of our cannonball. Keep in mind that components that are zero must be carried along, as well: 𝑎 = 0  ,−𝑔 would be the proper way to express the acceleration, in our projectile problem. Although this notation is more compact (which is why it’s used in computer algorithms), it can be cumbersome when performing subsequent math, since you have to carefully track the ordering of terms every time you perform an operation. That being the case, you will probably only want to use this notation at the very start or very end of a problem—in between, a mix of notations #1 and #3 would probably be most efficient.