Vectors
Article objectives
Vector Notation
The idea of components freed us from the confines of one-dimensional physics, but the component notation can be unwieldy, since every one-dimensional equation has to be written as a set of three separate equations in the three-dimensional case. Newton was stuck with the component notation until the day he died, but eventually someone figured out a way of abbreviating three equations as one.
(a) \(\overrightarrow{F}\)\(_{A}\)on B=-\(\overrightarrow{F}\)\(_{B}\) on A stands for: \(F_{A}\) on \(B_{x}\)=-\(F_{B}\) on \(A_{x}\); \(F_{A}\) on \(B_{y}\)=-\(F_{B}\) on \(A_{y}\); \(F_{A}\) on \(B_{z}\)=-\(F_{B}\) on \(A_{z}\)
(b) \(\overrightarrow{F}\)\(_{total}\)=\(\overrightarrow{F}\)\(_{1}\)+\(\overrightarrow{F}\)\(_{2}\)+… stands for: \(F_{total,x}\)=\(F_{1,x}\)+\(F_{2,x}\)+… \(F_{total,y}\)=\(F_{1,y}\)+\(F_{2,y}\)+… \(F_{total,z}\)=\(F_{1,z}\)+\(F_{2,z}\)+…
(c) \(\overrightarrow{a}\)=Δ\(\overrightarrow{v}\)Δt stands for: \(a_{x}=Δv_{x}/Δt\) \(a_{y}=Δv_{y}/Δt\) \(a_{z}=Δv_{z}/Δt\)
Example (a) above shows both ways of writing Newton's third law.
The idea is that each of the algebra symbols with an arrow written on top, called a vector, is actually an abbreviation for three different numbers, the x, y, and z components. The three components are referred to as the components of the vector, e.g., \(F_{x}\) is the x component of the vector \(\overrightarrow{F}\) . The notation with an arrow on top is good for handwritten equations, but is unattractive in a printed book, so books use boldface, F, to represent vectors.
In general, the vector notation is useful for any quantity that has both an amount and a direction in space. Even when you are not going to write any actual vector notation, the concept itself is a useful one. We say that force and velocity, for example, are vectors. A quantity that has no direction in space, such as mass or time, is called a scalar. The amount of a vector quantity is called its magnitude. The notation for the magnitude of a vector A is |A| , like the absolute value sign used with scalars.
Often, as in example (b), we wish to use the vector notation to represent adding up all the x components to get a total x component, etc. The plus sign is used between two vectors to indicate this type of component-by-component addition. Of course, vectors are really triplets of numbers, not numbers, so this is not the same as the use of the plus sign with individual numbers. But since we don't want to have to invent new words and symbols for this operation on vectors, we use the same old plus sign, and the same old addition-related words like “add,” “sum,” and “total.” Combining vectors this way is called vector addition.
Similarly, the minus sign in example (a) was used to indicate negating each of the vector's three components individually. The equals sign is used to mean that all three components of the vector on the left side of an equation are the same as the corresponding components on the right.
Example (c) shows how we abuse the division symbol in a similar manner. When we write the vector Δv divided by the scalar Δt, we mean the new vector formed by dividing each one of the velocity components by Δt.
It's not hard to imagine a variety of operations that would combine vectors with vectors or vectors with scalars, but only four of them are required in order to express Newton's laws:
| Operation | Definition |
|---|---|
| vector+vector | Add component by component to make a new set of three numbers. |
| vector-vector | Subtract component by component to make a new set of three numbers. |
| vector⋅scalar | Multiply each component of the vector by the scalar. |
| vector/scalar | Divide each component of the vector by the scalar. |
As an example of an operation that is not useful for physics, there just aren't any useful physics applications for dividing a vector by another vector component by component.
We can do algebra with vectors, or with a mixture of vectors and scalars in the same equation. Basically all the normal rules of algebra apply, but if you're not sure if a certain step is valid, you should simply translate it into three component-based equations and see if it works.
Example 1: Order of addition
◊ If we are adding two force vectors, F+G , is it valid to assume as in ordinary algebra that F+G is the same as G+F ?
◊ To tell if this algebra rule also applies to vectors, we simply translate the vector notation into ordinary algebra notation. In terms of ordinary numbers, the components of the vector F+G would be \(F_{x}+G_{x}, F_{y}+G_{y}\), and \(F_{z}+G_{z}\), which are certainly the same three numbers as \(G_x+F_x, G_y+F_y,\) and \(G_z+F_z\). Yes, F+G is the same as G+F .
It is useful to define a symbol r for the vector whose components are x, y, and z, and a symbol Δr made out of Δ x, Δ y, and Δ z.
Although this may all seem a little formidable, keep in mind that it amounts to nothing more than a way of abbreviating equations!
Figure a: The x and y components of a vector can be thought of as the shadows it casts onto the x and y axes.
Figure b: Example vector with an x and y component drawn as an arrow.
Drawing vectors as arrows
A vector in two dimensions can be easily visualized by drawing an arrow whose length represents its magnitude and whose direction represents its direction. The x component of a vector can then be visualized as the length of the shadow it would cast in a beam of light projected onto the x axis, and similarly for the y component. Shadows with arrowheads pointing back against the direction of the positive axis correspond to negative components.
In this type of diagram, the negative of a vector is the vector with the same magnitude but in the opposite direction. Multiplying a vector by a scalar is represented by lengthening the arrow by that factor, and similarly for division.
Calculations with magnitude and direction
If you ask someone where Las Vegas is compared to Los Angeles, they are unlikely to say that the Δ x is 290 km and the Δ y is 230 km, in a coordinate system where the positive x axis is east and the y axis points north. They will probably say instead that it's 370 km to the northeast. If they were being precise, they might give the direction as 38° counterclockwise from east. In two dimensions, we can always specify a vector's direction like this, using a single angle. A magnitude plus an angle suffice to specify everything about the vector. The following two examples show how we use trigonometry and the Pythagorean theorem to go back and forth between the x-y and magnitude-angle descriptions of vectors.
Figure c: Example 2 and 3
The following example shows the correct handling of the plus and minus signs, which is usually the main cause of mistakes.
Example 4: Negative components
◊ San Diego is 120 km east and 150 km south of Los Angeles. An airplane pilot is setting course from San Diego to Los Angeles. At what angle should she set her course, measured counterclockwise from east, as shown in the figure?
◊ If we make the traditional choice of coordinate axes, with x pointing to the right and y pointing up on the map, then her Δ x is negative, because her final x value is less than her initial x value. Her Δ y is positive, so we have
$$Δx =-120 km $$ $$Δy =150 km.$$ If we work by analogy with example 2, we get
$$θ = tan^{-1} \frac{\Delta{y}}{\Delta{x}}$$ $$=tan^{-1} (-1.25)$$ $$=-51^\circ$$
According to the usual way of defining angles in trigonometry, a negative result means an angle that lies clockwise from the x axis, which would have her heading for the Baja California. What went wrong? The answer is that when you ask your calculator to take the arctangent of a number, there are always two valid possibilities differing by 180°. That is, there are two possible angles whose tangents equal -1.25:
$$tan 129^\circ = -1.25$$ $$tan -51^\circ = -1.25$$
You calculator doesn't know which is the correct one, so it just picks one. In this case, the one it picked was the wrong one, and it was up to you to add 180° to it to find the right answer.
Example 5: A shortcut
◊ A split second after nine o'clock, the hour hand on a clock dial has moved clockwise past the nine-o'clock position by some imperceptibly small angle φ. Let positive x be to the right and positive y up. If the hand, with length ℓ, is represented by a Δr vector going from the dial's center to the tip of the hand, find this vector's Δ x.
◊ The following shortcut is the easiest way to work out examples like these, in which a vector's direction is known relative to one of the axes. We can tell that Δr will have a large, negative x component and a small, positive y. Since Δ x<0, there are really only two logical possibilities: either Δ x = -ℓ cosφ, or Δ x = -ℓ sinφ. Because φ is small, cosφ is large and sinφ is small. We conclude that Δ x = -ℓ cosφ.
A typical application of this technique to force vectors is given in example 6.
Techniques for adding vectors
Addition of vectors given their components
The easiest type of vector addition is when you are in possession of the components, and want to find the components of their sum.
Example 6: Adding components ◊ Given the Δ x and Δ y values from the previous examples, find the Δ x and Δ y from San Diego to Las Vegas.
◊
$$Δx_{total} =Δx_1+Δx_2 =-120 km+290 km =170 km $$ $$Δy_{total} =Δy_1+Δy_2 =150 km+230 km =380$$
Note how the signs of the x components take care of the westward and eastward motions, which partially cancel.
Addition of vectors given their magnitudes and directions
In this case, you must first translate the magnitudes and directions into components, and then add the components. In our San Diego-Los Angeles-Las Vegas example, we can simply string together the preceding examples.
Graphical addition of vectors
Figure f: Vectors can be added graphically by placing them tip to tail, and then drawing a vector from the tail of the first vector to the tip of the second vector.
Often the easiest way to add vectors is by making a scale drawing on a piece of paper. This is known as graphical addition, as opposed to the analytic techniques discussed previously. (It has nothing to do with x-y graphs or graph paper. “Graphical” here simply means drawing. It comes from the Greek verb “grapho,” to write, like related English words including “graphic.”)
Example 7: LA to Vegas, graphically
◊ Given the magnitudes and angles of the Δr vectors from San Diego to Los Angeles and from Los Angeles to Las Vegas, find the magnitude and angle of the Δr vector from San Diego to Las Vegas.
◊ Using a protractor and a ruler, we make a careful scale drawing, as shown in figure g. The protractor can be conveniently aligned with the blue rules on the notebook paper. A scale of 1 mm → 2 km was chosen for this solution because it was as big as possible (for accuracy) without being so big that the drawing wouldn't fit on the page. With a ruler, we measure the distance from San Diego to Las Vegas to be 206 mm, which corresponds to 412 km. With a protractor, we measure the angle θ to be \(65^\circ{}\).
Even when we don't intend to do an actual graphical calculation with a ruler and protractor, it can be convenient to diagram the addition of vectors in this way. With Δr vectors, it intuitively makes sense to lay the vectors tip-to-tail and draw the sum vector from the tail of the first vector to the tip of the second vector. We can do the same when adding other vectors such as force vectors.
Unit vector notation
When we want to specify a vector by its components, it can be cumbersome to have to write the algebra symbol for each component:
$$Δ x= 290 km, Δ y=230 km$$
A more compact notation is to write
$$Δr=(290 km)\hat{x}+(230 km)\hat{y},$$
where the vectors \(\hat{x} , \hat{y}\) , and \(\hat{z}\) , called the unit vectors, are defined as the vectors that have magnitude equal to 1 and directions lying along the x, y, and z axes. In speech, they are referred to as “x-hat” and so on.
A slightly different, and harder to remember, version of this notation is unfortunately more prevalent. In this version, the unit vectors are called\(\hat{i} , \hat{j}\) , and \(\hat{k}\):
$$Δr=(290 km)\hat{i}+(230 km)\hat{j}.$$
Rotational invariance
Figure h: Component-by-component multiplication of the vectors in 1 would produce different vectors in coordinate systems 2 and 3.
Let's take a closer look at why certain vector operations are useful and others are not. Consider the operation of multiplying two vectors component by component to produce a third vector:
$$R_x =P_x Q_x $$ $$R_y =P_y Q_y$$ $$ R_z =P_z Q_z$$
As a simple example, we choose vectors P and Q to have length 1, and make them perpendicular to each other, as shown in figure h/1. If we compute the result of our new vector operation using the coordinate system in h/2, we find:
$$R_x =0$$ $$ R_y =0$$ $$ R_z =0$$
The x component is zero because \(P_x\)=0, the y component is zero because \(Q_y\)=0, and the z component is of course zero because both vectors are in the x-y plane. However, if we carry out the same operations in coordinate system h/3, rotated 45 degrees with respect to the previous one, we find
$$R_x =1/2$$ $$R_y =-1/2$$ $$R_z =0$$
The operation's result depends on what coordinate system we use, and since the two versions of R have different lengths (one being zero and the other nonzero), they don't just represent the same answer expressed in two different coordinate systems. Such an operation will never be useful in physics, because experiments show physics works the same regardless of which way we orient the laboratory building! The useful vector operations, such as addition and scalar multiplication, are rotationally invariant, i.e., come out the same regardless of the orientation of the coordinate system.
Some smart phones and GPS units contain electronic compasses that can sense the direction of the earth's magnetic field vector, notated B. Because all vectors work according to the same rules, you don't need to know anything special about magnetism in order to understand this example. Unlike a traditional compass that uses a magnetized needle on a bearing, an electronic compass has no moving parts. It contains two sensors oriented perpendicular to one another, and each sensor is only sensitive to the component of the earth's field that lies along its own axis. Because a choice of coordinates is arbitrary, we can take one of these sensors as defining the x axis and the other the y. Given the two components \(B_{x}\) and \(B_{y}\), the device's computer chip can compute the angle of magnetic north relative to its sensors, \(tan^{-1} B_{y}\) / \(B_{x}\) .
All compasses are vulnerable to errors because of nearby magnetic materials, and in particular it may happen that some part of the compass's own housing becomes magnetized. In an electronic compass, rotational invariance provides a convenient way of calibrating away such effects by having the user rotate the device in a horizontal circle.
Suppose that when the compass is oriented in a certain way, it measures \(B_{x}\) =1.00 and \(B_{y}\) =0.00 (in certain units). We then expect that when it is rotated 90 degrees clockwise, the sensors will detect \(B_{x}\) =0.00 and \(B_{y}\) =1.00.
But imagine instead that we get \(B_{x}\) =0.20 and \(B_{y}\) =0.80. This would violate rotational invariance, since rotating the coordinate system is supposed to give a different description of the same vector. The magnitude appears to have changed from 1.00 to 0.202+0.802=0.82 , and a vector can't change its magnitude just because you rotate it. The compass's computer chip figures out that some effect, possibly a slight magnetization of its housing, must be adding an erroneous 0.2 units to all the \(B_{x}\) readings, because subtracting this amount from all the \(B_{x}\) values gives vectors that have the same magnitude, satisfying rotational invariance.