Anderson Video - Adding Vector Components Example

Professor Anderson
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Hello class, Professor Anderson here. Let's take a look at an example problem this is one that we already talked about but let's take a look at it again with this idea of breaking the vectors into components in terms of these unit vectors and then adding them up as vectors. Okay. So let's review our picture. We're gonna run 1 mile to the north, then 0.4 miles west and then 0.1 mile south. Okay. So our first leg of our journey is straight up one mile and then we go zero point four miles to the west and then we go zero point one miles south. And we want to figure out this guy right here. What is that R vector? Once you identify the vector you know everything about it. Okay? You know the length and you can calculate the angle. So all we need is the vector and let's label these A, B, and C, and let's figure out how to write these things in terms of the unit vectors that we just talked about. Okay? So this one, this first leg that's vector A. How do I write that as a vector in this XY coordinate space? I need something multiplying i-hat and then I need something multiplying j-hat. i-hat is in the X direction in this case it would be along the east axis. So, what do I put there? What should I put there? Yeah. >> (student speaking) Um. Zero. >> Zero. All right. This first one doesn't go any in the East. Okay. What do I need to put on the second term? One. Right? One mile. So this is how I write that first vector it's zero i-hat, plus 1 j-hat. The second vector, B, is gonna be what? Well we've got something in front of the i-hat. We have something in front of the j-hat. What do you think? Yeah. >> (student speaking) I'm confused as how- where did the zero come from. >> Okay, good question. So, this East axis is what we're gonna say is our typical x axis. Which means that i-hat points along that axis. Okay? So for the first vector, vector A, was it going at all in the i-hat direction? It was going exactly zero in the I hat direction. Okay. So we don't technically have to write that we just cross it out entirely but we're trying to be clear here. Alright, and of course it went 1 in the other direction, one mile in the other direction which is j-hat. What about B? Let's do the easy part first what should I put there on the j-hat for vector B? How far north is it going? Was that a okay or was that a zero? That's a zero. That's a zero, right? Vector B is going perfectly West it's not going at all north. What do we put on the first one? You're itching to raise your hand and shout something out, go for it. >> (student speaking) It should be 0.45. I don't know if it's negative. Is it supposed to be negative? >> All right. Let's do the easy part first 0.4. And now the question is it positive or is it negative? Alright. When you are moving along the x axis, its positive. If I'm moving along the negative x axis, then it is negative. So is that thing moving to the right or to the left? >> (student speaking) To the left. It's moving to the left, and so we want to make this negative. Good. All right, almost there. Finally we got to do vector C. What is vector C? Well we know that it's not moving in the east or west, so that first term is zero. And now based on what we just did, what should I put there? You guys know the answer to this it's negative 0.1 Okay? All right. Those are our three vectors written out in this component form and now it's nearly trivial to write the resultant vector. How do I do it? I add these up. All the i-hats I add up. I add 0 to negative 0.4 to 0 and of course I get negative 0.4. And now I add up these 3. I have 1 plus 0 plus negative 0.1. And that gives me a positive 0.9 j-hat. And now you're essentially done. Once you know the vector you essentially know everything about the vector. Right? Calculating the magnitude is just a matter of squaring the first term, squaring the second term, and taking the square root and we already did this we know what the answer is it's 0.98. And if you want to figure out say this angle right here, phi, how do we do that? Well let's draw the triangle phi this is 0.98 which we said was R. This was 0.9 which that's going to be our parse of Y and this is our R sub X which was negative 0.4. Okay. Those are the three sides of our triangle this is again a right angle and so phi you can do any trig relation you like, but let's take the arc cosine of 0.9 over 0.98, and if I remember right I think we got 23 degrees. Okay? Any questions about this one? Any questions about how we approached it with the unit vectors? Okay, good. Hopefully that one is clear, if not come see me an office hours. Cheers.
Hello class, Professor Anderson here. Let's take a look at an example problem this is one that we already talked about but let's take a look at it again with this idea of breaking the vectors into components in terms of these unit vectors and then adding them up as vectors. Okay. So let's review our picture. We're gonna run 1 mile to the north, then 0.4 miles west and then 0.1 mile south. Okay. So our first leg of our journey is straight up one mile and then we go zero point four miles to the west and then we go zero point one miles south. And we want to figure out this guy right here. What is that R vector? Once you identify the vector you know everything about it. Okay? You know the length and you can calculate the angle. So all we need is the vector and let's label these A, B, and C, and let's figure out how to write these things in terms of the unit vectors that we just talked about. Okay? So this one, this first leg that's vector A. How do I write that as a vector in this XY coordinate space? I need something multiplying i-hat and then I need something multiplying j-hat. i-hat is in the X direction in this case it would be along the east axis. So, what do I put there? What should I put there? Yeah. >> (student speaking) Um. Zero. >> Zero. All right. This first one doesn't go any in the East. Okay. What do I need to put on the second term? One. Right? One mile. So this is how I write that first vector it's zero i-hat, plus 1 j-hat. The second vector, B, is gonna be what? Well we've got something in front of the i-hat. We have something in front of the j-hat. What do you think? Yeah. >> (student speaking) I'm confused as how- where did the zero come from. >> Okay, good question. So, this East axis is what we're gonna say is our typical x axis. Which means that i-hat points along that axis. Okay? So for the first vector, vector A, was it going at all in the i-hat direction? It was going exactly zero in the I hat direction. Okay. So we don't technically have to write that we just cross it out entirely but we're trying to be clear here. Alright, and of course it went 1 in the other direction, one mile in the other direction which is j-hat. What about B? Let's do the easy part first what should I put there on the j-hat for vector B? How far north is it going? Was that a okay or was that a zero? That's a zero. That's a zero, right? Vector B is going perfectly West it's not going at all north. What do we put on the first one? You're itching to raise your hand and shout something out, go for it. >> (student speaking) It should be 0.45. I don't know if it's negative. Is it supposed to be negative? >> All right. Let's do the easy part first 0.4. And now the question is it positive or is it negative? Alright. When you are moving along the x axis, its positive. If I'm moving along the negative x axis, then it is negative. So is that thing moving to the right or to the left? >> (student speaking) To the left. It's moving to the left, and so we want to make this negative. Good. All right, almost there. Finally we got to do vector C. What is vector C? Well we know that it's not moving in the east or west, so that first term is zero. And now based on what we just did, what should I put there? You guys know the answer to this it's negative 0.1 Okay? All right. Those are our three vectors written out in this component form and now it's nearly trivial to write the resultant vector. How do I do it? I add these up. All the i-hats I add up. I add 0 to negative 0.4 to 0 and of course I get negative 0.4. And now I add up these 3. I have 1 plus 0 plus negative 0.1. And that gives me a positive 0.9 j-hat. And now you're essentially done. Once you know the vector you essentially know everything about the vector. Right? Calculating the magnitude is just a matter of squaring the first term, squaring the second term, and taking the square root and we already did this we know what the answer is it's 0.98. And if you want to figure out say this angle right here, phi, how do we do that? Well let's draw the triangle phi this is 0.98 which we said was R. This was 0.9 which that's going to be our parse of Y and this is our R sub X which was negative 0.4. Okay. Those are the three sides of our triangle this is again a right angle and so phi you can do any trig relation you like, but let's take the arc cosine of 0.9 over 0.98, and if I remember right I think we got 23 degrees. Okay? Any questions about this one? Any questions about how we approached it with the unit vectors? Okay, good. Hopefully that one is clear, if not come see me an office hours. Cheers.