Episode 073: Dot And Cross Product Examples Pt. 1

Download this episode (right click and save)

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.

2-D Cross Product Question: For the following vectors c = [2,5] and d = [-4,9] calculate c x d and calculate d x c. First, let’s calculate c x d:

Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [2,5] and [-4,9]. For later reference, let’s identify each component:

ax = 2

ay = 5

bx = -4

by = 9

Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:

ax * by = 2 * 9 = 18

Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:

ay * bx = 5 * -4 = -20

Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:

18 - -20 = 38

Step 5: Place a k-hat at the end and box your answer, you’re done! So our k-hat represents that our vector is pointing entirely in the z-direction. So our answer could be written 38*k-hat. Alternatively, we can write this

c x d = [0,0,38].

Next, let’s calculate the opposite, d x c.

Step 1: Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]- These are given in the problem as [-4,9] and [2,5]. Notice that these are OPPOSITE from last time. For later reference, let’s identify each component:

ax = -4

ay = 9

bx = 2

by = 5

Step 2: Multiply ax and by- Here we can easily reference step #1 and perform the following operation:

ax * by = -4 * 5 = -20

Step 3: Multiply ay and bx- Here again we can easily reference step #1 and perform the following operation:

ay * bx =9 * 2 = 18

Step 4: Subtract step 2 from step 3, i.e. ax*by-ay*bx:

-20 - 18 = -38

Step 5: Place a k-hat at the end and box your answer, you’re done! So our k-hat represents that our vector is pointing entirely in the z-direction. So our answer could be written -38*k-hat. Alternatively, we can write this

d x c = [0,0,-38].

What do you notice between these two different operations?

What is different?

The vector is pointing in the opposite direction (change of sign)

What is similar?

The magnitude.

The entire vector is directed in the z-direction (i.e. x and y components are 0)

What can we draw from this?

One thing that we can notice, is that

c x d = -(d x c)

Is it possible to prove this?

Episode 072: The Right Hand Rule.

Download this episode (right click and save)

Before we dive into the examples, there is one last item. Throughout the coordinate systems and vector properties/components sections, have you wondered how the x, y, and z-coordinates are related? They are related by something known as the right hand rule. With your right hand, point your fingers flat out. If you were to point your fingers towards the along the x-axis, and then curl your fingers towards the y-axis, your thumb will point in the direction of the z-axis. This is something that you will see all the time if you study electricity and magnetism. Current, electric, and magnetic fields all relate in direction by the right hand rule.

Cross products follow the right hand rule: i-hat cross j-hat is k-hat. Of course, i-hat is the unit vector pointing in the x-direction, j-hat aligns with the y-direction and k-hat the z-direction. Following the axes, j-hat cross k-hat is i-hat. You could use your right hand rule to prove this.

Here is a trick: Think of the letters ijkijk. If you run to the right, any combination of three letters yields a positive cross product, i x j = k, j x k = i, k x i = j. If you run the letters to the left, any combination of three letters yields a negative cross product: k x j = -i, j x i = -k, and so on. This can give you a general idea of which direction your cross product vector is pointing. Now, on with the examples.

If you'd like to support the show, head on over to the app store and grab an app. It'll help the show, and it'll help you on your journey to mastering physics and mathematics! Great reviews for before a test, quiz, or final! Grab them all here or at the top of the page.

Episode 071: The Cross Product in 2 & 3 Dimensions!

Download this episode (right click and save)

Like the dot product, the vector cross product is another means of vector multiplication. The difference with the cross product being that the operation yields a vector quantity. To be honest, I had some hesitation including this in the bootcamp. It’s one of those things I honestly don’t think you’ll use incredibly often. However, if needed, it’s a very handy tool, and if you’re in a class where you do need it, it’s best to have seen it at least once before. So, without further ado, here are the 5 steps to performing the cross product operation in 2 Dimensions:

1) Identify Two Vectors in Coordinate System as [ax,ay] & [bx,by]

2) Multiply ax and by

3) Multiply ay and bx

4) Subtract step 2 from step 3, i.e. ax*by-ay*bx

5) Place a k-hat at the end and box your answer, you’re done!

What the heck k-hat? In 2-dimensions, the unit vectors which point in the “x” and “y” directions are called i-hat and j-hat. k-hat points along the z-direction in a 3-D coordinate system and has a length of 1 unit. This gives the vector its direction. When given two vectors, the cross product vector will always point perpendicular to both vectors. More on this shortly. But, it is possible to write your cross product vector as [0,0,ax*by-ay*bx]. So, the cross product kind of forces you to be in 3 dimensions whether you like it or not.

So if that’s the case, how does it work in 3 dimensions? Different steps, and 6 steps this time:

1) Identify Two Vectors in Coordinate System as [ax,ay,az] & [bx,by,bz]

2) Calculate [ay*bz-by*az]*i-hat

3) Calculate [az*bx-ax*bz]*j-hat

4) Calculate [ax*by-ay*bx]*k-hat

5) Write your vector as a vector consisting of the last 3 steps: [step2,step3,step4]

6) Box your answer, you’re done!

So, these are the steps to follow when calculating a cross product in 2 or 3 dimensions. Actually performing examples will make this process clearer and more fluid.

Episode 070: The Determinant.

Download this episode (right click and save)

This week, I decided to go back to a topic that was in-progress and sort of went on hold: vector operations. On the heels of the dot product, we're at the point where it's time to tackle the big one: the cross product. But first, there is a small bit of back story that is worth telling in order for the cross product to make sense: The Determinant.

So without going into too much painful detail, there is a wonderful subsection of mathematics called Elementary Linear Algebra, which is pretty much entirely dedicated to matrix operations. Fascinating course. So a matrix, since we haven't already covered it in any detail here on the blog/podcast you can think of as a square or rectangular array of numbers. Below is an example of a 2x2 matrix (pronounced 2 by 2):

You can put anything in these slots 1,2,3,4. Numbers, variables, trig functions, anything mathematical that you can think of. However, our focus right now is how to calculate the determinant. You can do this by multiplying the items in slots 1 and 4, and subtracting the product of slots 3 and 2.

Now, you can in the same way visualize a 3x3 matrix by thinking of each entry as one of the numbers on your phone.

In slots 1,2, &3, we will be putting our i-hat, j-hat, and k-hat unit vectors. To perform the first determinant, which will be for the i-hat vector, we will think of our lowest right-hand corner 2x2 block to be a 2x2 matrix, for which we will be calculating the determinant:

When you calculate the i-hat component of your determinant, we will draw one line horizontal, and one line vertically which will "cancel" out or show us which values we will not be using to calculate our determinant. This leaves (5*9-8*7)i-hat.

For the k-hat component, we perform roughly the same operation. The two pink lines show which components of the matrix we will not be using, and we calculate the determinant of the 2x2 matrix which remains: (4*8-7*5)k-hat.

Wait! You forgot about j-hat! What gives?? i-hat and k-hat are very similar in that they give "nice" 2x2 matrices to calculate determinants of. When we draw our pink lines again for this example:

We have a small problem. We have to cross over the pink lines! There is not an easy 2x2 matrix to calculate the determinant of! Well, if you were to move reproduce the i-hat column to the right of the k-hat column, there would be a nice 2x2 matrix to calculate. However, so as not to move the matrix around, there is a rule: when you cross the pink line for j-hat, you need to multiply the whole thing by -1: this gives -(4*9-7*6)j-hat, or (7*6-4*9) j-hat.

Altogether, we now have: (5*9-8*7)i-hat+(7*6-4*9)j-hat+(4*8-7*5)k-hat. Written as a vector where i-hat, j-hat and k-hat correspond to x,y,z coordinates we have [(5*9-8*7),(7*6-4*9),(4*8-7*5)]. Bear in mind, that these are not the numbers 1,2,3,4,5,6,7,8 & 9. They correspond to whatever variables, constants, or trig functions that rest in these spaces. More on this later. This whole operation may be marginally clumsy and a little confusing for newcomers, but stay tuned for the cross product step-by-step. The steps will be very similar, and this step in the journey will make the cross product more familiar, and seriously clarify its origins.

Episode 069: The Physics Quiz - Particles

Download this episode (right click and save)

First off, Happy Autumnal Equinox! The first day of fall is upon us! At the equator, the length of the day and night is roughly equal, hence the name equi, meaning equal, and nox meaning night.

This week, the podcast episode focuses on an app that I came across a couple of weeks ago, called The Physics Quiz. I really like this app. It contains over 600 questions relating to particle physics created by Dr. P. Nilsson, a scientist stationed at CERN, a particle accelerator facility located in Geneva Switzerland. I reached out to him regarding featuring his app on the show and got his blessing. It was very exciting talking with someone who does research at one of the leading science facilities in the world!

I took the quiz 3 times, and included on the podcast is me on take 3, on the easy setting. It's a fun quiz. On take 3 I scored an 8 out of 10. I'll be posting take 1 and take 2 as bonus Physics Phrydays.

I think apps like this are fantastic. They test your knowledge and keep you interested, and make physics not seem like work. Its something that you can play with friends who aren't even in the physics field. I without warning sprung a quiz on one of my friends while we were out at dinner. It sparked a lot of interesting conversation and we had a good time!

One word of warning: Quiz levels Medium and Hard are rough. I scored a 6/10 on medium and 4/10 on hard. The questions are excellent and help you learn, but you are certainly going to have to earn your score! Another great thing about this app: the price. It's only 99 cents! Well worth the dollar and certainly has earned a permanent place on my iPhone. I find myself coming to it when I have a few minutes of downtime here and there. Who knows, maybe I'll spring it on some future guests of the show!

You can reach out to The Physics Quiz on twitter Here or click the link at the top to check it out for yourself in the app store. Enjoy!

Physics Phrydays 05: Ten Physics Questions You Should Know as a Physics Graduate!

Physics Phrydays returns! I have been feeling very pop-quizzy the last week or so, so I decided to bring back Physics Phrydays because I found this really cool quiz online, and thought it would be fun to take it live in real time and share it with you. (I got 9/10, not bad!). Check out the quiz HERE if you want to play along.

This episode has a two pronged effect. 1) I am preparing for a very different type of quiz coming up on the next episode of the podcast. To test my strength and knowledge, I thought I'd ease in with a softball quiz and post it for your listening pleasure. It's actually surprising how much of it we've covered so far on the podcast. Allegdly, these are questions you should be able to answer as a Physics graduate, so keep it in mind. If you're a graduate, do you remember all of this stuff?

Second part of the two pronged effect: I am still committed to finishing The Elegant Universe, and so I'm trying to slowly bring back Physics Phrydays, and encourage myself into reading that book. It has again eluded me. But, here we go, slow but measured progress. Enjoy!

Episode 068: 7 Question Physics Brush Up Pop Quiz!

Download this episode (right click and save)

Pop Quiz:

1) What is the derivative of cos(x)?

2) What is the 45 degrees in radians?

3) Remember the cosine of 45 degrees?

4) Can you convert kilometers to miles?

5) Which two sides of a right triangle are used to calculate tangent?

6) Is displacement a vector or a scalar?

7) An object is launched into the air at an angle of 60 degress with respect to the horizontal. What are the velocities in the x and y directions?

--------------------------------------- ANSWER ALERT!!!
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
V

So, how did you do??? Here are the answers:

1) What is the derivative of cos(x)?: The answer is -sin(x). We haven't quite covered it here yet, but the derivatives of trig functions are essential for doing well in Physics.

2) What is the 45 degrees in radians?: The answer is pi/4. We can calculate this by multiplying 45 * pi/180 = pi/4.

3) Remember the cosine of 45 degrees?: The answer is sqrt(2)/2. We can remember this by reviewing Episode 24: Sines and Cosines by Counting to 4 episode of the podcast.

4) Can you convert kilometers to miles?: 1km = 0.621mi. If you know that there are roughly 1.6 km in a mile, we can use a conversion factor to solve: 1 km * 1 mi/1.6 km = 0.621mi.

5) Which two sides of a right triangle are used to calculate tangent?: Recalling sohcahtoa, we can use TOA, or tangent = opposite/adjacent to answer this question.

6) Is displacement a vector or a scalar? Displacement is a vector quantity, while distance is a vector, as it does not include a direction.

7) An object is launched into the air at an angle of 60 degress with respect to the horizontal. What are the velocities in the x and y directions?: This is a matter of breaking our velocity vector into components. We're not given a velocity so we can use a generic velocity v. vx = v*cos(60) = v/2 and vy = v*sin(60) = v*sqrt(3)/2.

So how did you do? This is a 7 question quiz. Each question was taken from a single app from the Physics Brush Up app bundle. Calculus. Radians vs. Degrees. Sin vs. Cosine. Unit Conversion. Sohcahtoa. Vectors vs. Scalars. Vector Components. If you were shaky on more than one question, you should consider grabbing it. You'll thank yourself later!