Physics 407, E. Möbius Section 3, Vectors

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3. Vectors

Viewgraph: “Map to Aletsch Glacier”Photo: “Up the mountain”

3.1 IntroductionSo far we have looked only at motion in one dimension. There are only two directions a

particle can take, which we take care of by assigning a plus or a minus sign to the magnitude ofthe displacement. In the real world – as far as we can see it – there is motion in three dimensions.Our path from the start to our final destination may at times be pretty elaborate. Even the pathon a map, with the distance to be walked, may not tell us the true story about the hike, unless weknow how to read also the more involved information, such as elevation lines. View MapThis aspect becomes quite obvious in a view of the mountain ahead. PhotoHowever, for many problems that we are going to solve in physics it is not necessary to knowthe exact path between start and destination. To locate the points and to compute, how muchenergy we need to get there, distance, direction and elevation may just be enough. Still as you cansee from the map and the photo, we need to know the distance and direction (with compass) onthe map and the direction (or elevation) upward. This sounds like three quantities. In fact we aretalking three dimensions here. In physics we often look for a way to simplify our task. To dothis for the current situation and many following in class, we use vectors.

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A vector has a magnitude as well as a direction. Depending on whether that is in a plane orin 3-D space, direction can have one angle or two angles to describe the direction. We will nowhow we have to work with vectors. We have introduced in our example a very specific vector, adisplacement vector, a vector that represents a displacement of an object. However, what wewill do now, will apply to all kinds of vectors (velocity, acceleration, etc.) in the same way lateron. The vector algebra and arithmetic, in this sense, is as widely applicable as ordinary algebraand arithmetic for simple magnitude quantities. We call them scalars. Not all quantities inphysics are vectors. There are many scalars as well as, such as temperature, given as a singlevalue with a sign.

Let us work a little more on displacement vectors. (Fig. 3-1)(Sketch on blackboard) If a particle moves from A to B, no matteralong which path, its actual displacement is represented by anarrow pointing from A to B. This is the graphic representation ofthe displacement vector. In books, vectors can be written as aletter with an arrowhead, an arrow underneath, or simply as boldface letter. Our course book uses the arrowhead that I will also useconsistently. Please interrupt me, if I don’t follow that rule. In thesame way as the value ∆x does not depend on the starting point, avector remains the same, if only the starting point is in differentpositions, but magnitude and direction remain the same. I.e. avector can be shifted in space, without changing its value. A vectorgives us only the net effect of a displacement. For most problemsin physics this is all that counts.

3.2 Adding VectorsTo add two displacement vectors means to evaluate the net

displacement after adding the two arrows head to tail, which isgiven by the vector equation:

r s = r a +

r b 3.2-1

The symbol + has a similar meaning as in ordinary algebra, butnote that it involves both magnitude and direction. The procedure to do this is most easily seengeometrically as adding the two arrows such that the new sum vector

r s extends from the tail of

r a to the head of

r b .

We can now easily see that it doesn’t matter in which sequence we add the two vectors.Demo Vector AddDemo Vector Add

The two ways of adding both vectors complete a parallelogram. This is the Commutative Lawof Vector Algebra:

r a +

r b =

r b + r a 3.2-2

By playing a little with our demonstration we see the role that the direction plays. If only themagnitudes were added, as for a scalar, we would get one unique new magnitude, i.e. themaximum that we can get out of this demonstration, if we stretch the gadget completely.Depending on the relative orientation of the two vectors the magnitude changes. We get theminimum out of this sum, if vector

r b points opposite to vector

r a .Next let us add three vectors

r a , r b and

r c . (Sketch on blackboard; Fig. 3-4 in book)The way in which we combine two vectors first and then add the third one, does not matter either.This is the Associative Law of Vector Algebra:

(r a +

r b ) +

r c = r a + (r b + r c ) 3.2-3

These laws look very similar to the equivalent ones in ordinary algebra. In the algebraic notation

Fig 3-1

Physics 407, E. Möbius Section 3, Vectors

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in formulae that is indeed the case. However, we must be reminded that always magnitude anddirection are involved.

The vector - r b has the same magnitude as

r b , but points in the opposite direction. Adding this

vector to r a has the effect of subtracting

r b , as we can also see in the two extreme cases of our

demo. Demo Vector AddDemo Vector Add

r d = r a -

r b = r a + (-

r b ) 3.2-4

The vector r d is the result of this subtraction. (Sketch on blackboard; Fig. 3-6 in book)

The general rules of algebra, how move terms from one side of the equation to the other, includingthe proper change in sign, apply also to vector algebra.

All these rules apply to vectors of any kind. However, only vectors of the same kind can beadded or subtracted. It makes no sense to add a displacement and a velocity, in the same way as youdon’t add a time to a distance.

3.3 Vector ComponentsAdding vectors geometrically does not tell us how to do the math with the vector addition. We

now need to learn how to calculate vector quantities mathematically. To do this properly we usea rectangular coordinate system. We will do this on the blackboard just in two dimensions, but itcan be naturally extended to three dimensions. View Computer(Fig. 3-8 from book) A component of a vector is the projection of the vector on one of the axes.We draw perpendicular lines to both axes from the start and end points of the vector to get the twocomponents, i.e. the vector is resolved into its components.

A component of a vector is a vector itself and has the same direction along the axis as theoriginal vector. In Fig. 3-8 both components point in the positive direction, i.e. are positive, while inthe example in (Fig. 3-9) the y-component is negative. View ComputerAgain, if we shift a vector without changing its dimensions, the components stay the same.

In Fig. 3-8 and Fig. 3-9 the vector direction is also given in an alternate way, as the angle Q. Wecan find the values for the components ax and ay of vector

r a from the right angle triangles by:ax = a cos Q ay = a sin Q 3.3-1

where a is the magnitude of the vector r a . Likewise we can reconstruct the vector by adding its

components head to tail.After resolving the vector into components the combination of them can be used instead of the

vector for computation. We see that vector r a is fully defined by a and Q. It is also determined

by ax and ay. They contain the same information. We can transform a vector from magnitude-anglenotation into component-notation and vice versa. Relations 3.3-1 give the transformation in onedirection, while:

a = ax2 + ay

2 and tanq =ay

ax

=a ⋅sinqa ⋅ cosq

3.3-2

provide the return transformation. In the case of three dimensions we need a magnitude and twoangles (a, Q, and F) or three components (ax, ay and az).

3.4 Unit VectorsA unit vector is defined as a vector with magnitude 1 (whatever the unit of that vector is; m,

m/s, m/s2 etc.) in a particular direction. It lacks both dimension and unit (these are left in themagnitude). It only gives us the particular direction. Henceforth, we will label the unit vectors inthe positive directions of the x, y and z-axes as ˆ i , ˆ j and ˆ k . The ^ sign is used instead of an arrowfor unit vectors. With these unit vectors we set up a right handed coordinate system (Fig. 3-14).It is called right-handed because it takes a counter-clockwise turn by 90o to turn the x into the y-axis and y into z. You can remember more easily how it is supposed to look by using your righthand. Spread thumb, digit and middle finger at right angles. Then the thumb is the x-axis, the digitthe y-axis, and the middle finger the z-axis. Here is another way to look at it as a right-handedcoordinate system.

Physics 407, E. Möbius Section 3, Vectors

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The unit vectors are useful to express other more general vectors such as:

r a = ax

ˆ i + ayˆ j 3.3-3

The quantitiesaxˆ i and ay

ˆ j are the vector components of r a and ax and ay are called the scalar

components of r a .They contain both the magnitude and units. We can write the displacement

vector of our hike from the lunch place to the overnight hut as:

r d = (-5km)ˆ i + (-2km) ˆ j + (+0.55km) ˆ k

i.e. about 5 km west ( ˆ i points east), 2 km south ( ˆ j points north), and 550 m up.

3.5 Adding Vectors by ComponentsNow we can repeat the adding of two vectors arithmetically. Let’s use

r s = r a +

r b 3.2-1

from before. In components this reads:sx = ax + bx

sy = ay + by 3.5-1sz = az + bz

Two vectors are equal, if their components are equal. This means, we can apply ordinary algebrato all scalar components, but we have to do it simultaneously as many times, as there arecomponents. Similarly, we can express this equation in unit vector notation to get the full vectorrepresentation.

3.6 Vectors and the Laws of PhysicsSo far we have used conveniently a coordinate system that aligns with the edges of the book orthe blackboard. However, if we are only interested in the physical vector quantities of adisplacement, velocity, acceleration etc. we don’t need to constrain ourselves in this way. This isa very important feature of physics, and in fact of nature itself, that important quantities andqualities of objects, motions and forces etc. remain identical or invariant under translation androtation. We call such quantities or parameters invariants. We will learn more about theconsequences of this later in the course. In particular, a vector remains the same, if we use adifferent coordinate system into which we resolve the vector. (Fig. 3-18) View ComputerSee also Fig 3-2 below.

In this new system the components are different, the angle is different in relation to the x-axis,but the magnitude remains the same and the direction in relation to any other object remains the

Fig. 3-2: The same Vector decomposed intocomponents in 2 different coordinate systems.

Physics 407, E. Möbius Section 3, Vectors

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same. If we just look at the vector, it hasn’t changed a bit. We just have changed our referenceframe. Which one of the systems is the right one? This is the wrong question. They are all valid.All produce the same magnitude and direction for the vector, and we get:

a = ax2 + ay

2 = a' x2 +a' y

2 and q =q ' +f 3.6-1The new system has been formed out of the old coordinate system by a rotation by the angle F.

This view is compatible with the fact that the physics is independent of the coordinate system, inwhich we attempt to describe it. This gives us great choice in the selection of coordinate systems,often to make our life simpler, when we try to solve a problem. For a motion along an inclinedplane, it may, for example, be convenient to choose a system that is aligned with the surface ofthe inclined plane rather than vertical and horizontal!

3.7 Multiplying VectorsAdding and subtracting vectors looks like a straight-forward extension of ordinary algebra.

However, this becomes much different when it comes to multiplication. However, we only needvector multiplication later on, starting with Chapter 7 and 12. Therefore, I will postpone theirtreatment until this point in the course.

At this point I only want to introduce the multiplication of a vector by a scalar s. If wemultiply vector

r a by s, its magnitude is increased by the factor s.

r d = r a ⋅ s = (ax ⋅ s)ˆ i + (ay ⋅ s) ˆ j + (az ⋅s) ˆ k and d = a ⋅ s q = q

Likewise, division by s reduces the magnitude by the same factor. For the components thismeans that each component must by multiplied or divided in the same way.