Episode 064: 5 Steps to Perform The Dot or Scalar Product!

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In the last section we covered vector addition. Adding two vectors will always yield a vector. With multiplication, it is perform two different operation. One of those operations yields a scalar. One yields a vector. In this section we will dive into the scalar product, otherwise known as the dot product. It is called the dot product because the symbol designating the scalar product is a dot. Written out, it looks something like this:

a = B ⋅ C

When not being denoted with an arrow, vectors are also commonly denoted in a bold font. Above, scalar a is equal to the scalar product between vectors B and C. The process of performing the dot product operation can be broken into 5 easy steps:

1) Identify two vectors- in this step you will need to simply locate the two vectors you’ll be “dotting”.

2) Identify number of dimensions- i.e. [2,0] and [4,3] are both two dimensional, because there are two numbers, or components, inside the brackets, [2,0,5] and [4,3,-9] are 3-dimensional vectors.

3) Multiply like components from each dimension together- Example: [bx,by,bz] and [cx,cy,cz], we now perform bx*cx, by*cy, and bz*cz.

4) Add Together All Component Multiples- For example, with our three dimensional vector: bx*cx+by*cy+bz*cz.

5) Box Your Answer, You’re Done!

a = B ⋅ C = bx*cx + by*cy + bz*cz

So this is a step-by-step skeleton to follow any time you need to perform a dot product operation, of which there will be no short supply during your 1st year of physics. Next, we will roll through one example just to understand exactly what’s going on with a little vector subtraction and then a little bit of dot product.

Question: A strapping young man exerted a force of [10N,10N,10N] moving a box from position (1,2,3) to position (4,5,6) on a 3-D cartesian axis, where all positions are defined in meters. What is the total work done?

Answer: Work is defined as follows:

W = F ⋅ d where W is the work (scalar), F is the force vector, and d is the displacement vector. In the problem, the displacement is not exactly given to us. For that, we need to know the following relationship: d = xf - xi where xf is the final position and xi is the initial position. First, since we know the initial and final displacements, let’s calculate the displacement vector: d = (4,5,6) - (1,2,3) d = [4-1,5-2,6-3] d = [3,3,3] So, our displacement vector is [3,3,3]. Notice in the vector subtraction that each component of the position was subtracted individually, and all three directions combined then give the final displacement vector. With both vectors in hand, we can now calculate the work: W = F ⋅ d = [10N,10N,10N] ⋅ [3m,3m,3m] (1) = 10N*3m + 10N*3m + 10N*3m (3) W = 30J + 30J + 30J (4) W = 90J (5) Where J stands for Joule, which is defined to be a N*m. Notice that the work could be the exact same if say the x-component of the dot product was 10 Joules more and the z-component 10 J less. Were you able to catch the steps in there? Each equation has a number in parentheses to the far right. These are the steps of dot product calculation. Notice that step (2) is missing. In the problem, we know that the coordinates are 3-dimensional. As you grow more comfortable with dot products, step (2) becomes more of a mental note than something that needs to be officially addressed.

Episode 063: 5 Critical Vector Properties for Components and the Importance of Coordinate Systems on Vector Operations

A little bit of an experiment here. Putting up a "rerun" episode. If you knew back then, what you know right now.... A little gauge to see how far you've come in the last 20 episodes. How much more comfortable are you with this topic now? Listen and find out!

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When considering how to introduce vectors as an introduction to vector components, I thought that it would be critical to dig into all the nitty gritty of vectors, dot products, cross products, unit vectors, which operations were commutative, etc. I decided that breaking vectors into components is a very useful operation that can be done in very few steps, with a relatively little amount of knowledge. The useful crux here is that you really don't need to have a depth of knowledge to understand what is going on here, be really good at breaking a vector into components, and using this technique all over your physics courses. I'm not saying that the other depth-stuff isn't important, but it's not critical to get the job done here.

1) Vectors can be imagined as an arrow of certain length. The length gives the magnitude, and the orientation of the arrow gives the direction.

2) Vectors slide!- Any vector of the same magnitude and direction is the same as any other in the coordinate system. You can slide vectors anywhere you'd like in the coordinate system and have the exact same properties! E.g. displacement. It doesn't matter where in the world you walk 5 miles, you still walked 5 miles.

3) Vectors can be represented as a single coordinate, [x,y,z]- Because vectors can slide, they can be described as a single point in the coordinate system, with the other point assumed to be the origin.

4) Since vectors can be described in terms of a coordinate, say, [x,y], the vector can be thought of as a hypotenuse, whose legs are the vector components which align with the x and y axis.

5) Vectors can add and subtract by adding and subtracting like components. Many applications, such as displacement, force, actually any vector fare, you will have to add or subtract vectors. The easiest way to do this is to add and subtract the x, y, and z components of each, and then recombine the resulting vector.

Do you see how all of the effort and emphasis on being fluent with sohcahtoa is going to come in handy? Those 6 steps are going to get embedded into a single step in the breaking vectors into components step-by-step. So, we should be applying our newly found sohcahtoa knowledge in no time!

Episode 062: Vector Operations pt 2- Vector Addition of 3 Distances

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Consider the next example. Standing back on your coordinate axis, you daringly decide to venture out, but not along any coordinate axis. First you walk roughly the same displacement as the last problem, to the point [4,3]. Next you continue on, 3m to the east and 4m to the north. Lastly, you take a few steps backwards, traveling 3m west and 1m north. What is your final location? We can see some illustration of this in the picture above. Each walk is shown in color, while the entire traversed distance is shown in black. Following the steps from before, each motion is laid from tip to tail, and the final displacement vector is drawn from the tail of the first vector to the tip of the final. If you had graph paper and performed this to scale, you could have an exact answer without having to do any sort of mathematics. The math side of things, we can simply add d1 to d2, and then add d3 to get our final vector.

d1 + d2 = [4,3] + [3,4] = [7,7]

(d1+d2)+d3 = [7,7] + [-3,1] = [4,8]

So, ultimately, on your little stroll, you walked 4m to the east, and 8m to the north. One thing that is incredibly illustrative about this example, but a little confusing is that in the walking example, the walker always ends at the same point that the next vector starts. In a physics problem, if you are given vectors in this way, and they are all centered around the origin, say in the case of a force problem, several forces are all acting on a single body at the origin, you can still add them in this way, because remember it is possible to slide vectors any which way, and as long as the magnitude and direction stays intact, the vector remains intact, so it is always possible to line them all tip to tail in this way, in any order, and get your resultant vector.

Episode 061: 8 Hot Podcast Episodes of the last 60...Your Picks and My Picks on a Happy Birthday!

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Happy 1 Year birthday to pwn your hw! Usually making it to the 10 episode mark is the first major podcast milestone, but the next is certainly the year, and the next the 100 ep mark, which we're looking right on track to clear. In this episode I wanted to take a moment to look backwards, and give a nod to the 5 most listened to podcasts of the year, as well as my top 3 picks. But first, the top 5 from the listeners:

5) Simulation Theory- BONUS Episode

4) Episode 043: 5 Steps to Break Any Vector Into Components (in Cartesian Coordinates) AND Combine Components Back Into The Original Vector!

3) Episode 035: sohcahtoa example #1- 3-4-5 triangle angle calculation-

2) Episode 004: Greek Letters-

1) Episode 026: 8 Ideas to Effectively and Productively Use Your Textbook.

So, the listeners have spoken. Those are the fav picks from the people who make the podcast happen. Next up, the 3 episodes of which on a look back, I took the biggest fancy to:

3) Episode 029: Wardenclyffe Tower, pt. 1

2) Episode 007: Powers of Ten

1) Episode 024: Sines and Cosines by Counting to 4!

The good news, we're picked up for another year, so looks like everything is going onwards, with a lot of new hot items on the stove. If you're feeling supportive, click the link on the top and grab yourself an app, it will make you a better person.

Episode 060- Coordinate Point Multiplication...How? Vector OperationsExplained -pt 1

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Now that we have some comfort with vectors, in being able to identify them, understanding some of their properties, and breaking them into components, there is one last large piece of the puzzle to having a solid foundation of vectors and vector operations. Because vectors are not simply one number, many of our mathematical tools do not apply. How do you add a coordinate point? How do you subtract it? Even worse, how do you multiply it? These answers will be found in this chapter, as well as some very useful vector properties, and their associated proofs. So let’s start with the usual first: addition. There are two methods of adding vectors, one mathematical, the other graphical. First the math: consider you have two vectors, [vx,vy,vz] and [wx,wy,wz]. The way to add these vectors to produce a vector is highly intuitive: add each component to create the component of your final vector, i.e. [vx+wx,vy+wy,vz+wz]. Finished. Nothing more to it. This actually gives us a very valuable result: vector addition is commutative, i.e. it does not matter which order you add the vectors. Likewise, subtraction gives a very similar result:

[vx,vy,vz]-[wx,wy,wz] = [vx-wx,vy-wy,vz-wz]

This, as you might expect, means that vector subtraction is not commutative, i.e. the order of vectors matters.

Consider a more visual representation of vector addition: Imagine that you are standing at the origin of a coordinate axis, like graph paper, and decide to go for a little walk. You first decide to tightrope your way down the positive x-axis and you move 4 meters. You then turn 90 degrees to your left and walk 3 meters up, parallel with the y-axis, but 5 meters displaced. What is your overall motion, or displacement? We’re considering two different displacement vectors in this scenario. The first is the tightrope down the x-axis, which moved us from position (0,0) to (4,0). This is equivalent to a displacement vector [4,0]. Next, the walk parallel with the y-axis. This is a displacement of [0,3]. The vector [0,3] is akin to simply having walked up the y-axis first. This is one of the wonderful properties of vectors. When you think of them as arrows, you can slide them around wherever you want, and they retain the same properties. That’s what we’re going to do here. So, our overall displacement will be the sum of each increment that we’ve walked. This means

[4,0] + [0,3] = [4+0,0+3] = [4,3]

In this example, it’s very easy to see how the addition works in each direction, as the motions are both conveniently along axes, giving us the easy task of adding an integer and zero.

Q: How come to calculate the displacement I don’t implement the Pythagorean Theorem?

A: If you wanted to calculate the magnitude of displacement, you would do exactly that, however the angle and magnitude of displacement are retained in the value [4,3]. You could also say that the overall displacement was 5m, at an angle of ~37 degrees w/r/t the x-axis. Mathematically, it is usually simpler to keep the entire displacement in terms of the overall x, y and sometimes z components. The takeaway from the example: in order to add vectors graphically,

1) “Lay” your first vector.

2) Place the “tail” of the second vector at the “tip” of the first.

3) Repeat for any following vectors that need to be added.

4) Connect the tail of the first vector to the tip of the last vector. This is your resultant vector.