Geometric Vectors - Video Tutorials & Practice Problems

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1

concept

Introduction to Vectors

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3m

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Welcome back everyone. So in this video, we're going to be talking about one of the more critical topics both in math, as well as future science courses you will likely take which is something known as vectors. Now to understand vectors, I want you to imagine that two people walk up to you. The first person tells you, hey, I just ran six MPH and the second person tells you, I just ran six MPH northeast. What is really the difference between what these two people told you? Well, the first person gave you a magnitude, they told you how much they were running. Now, the second person gave you a magnitude, but they also gave you a direction, they told you where they were running. And this second person gave you an example of a vector. A vector is a quantity that has both magnitude and direction. Now to understand mathematically how these vectors are represented and what they look like. I think it's best if we just jump into some examples of a vector. So vectors visually are represented by arrows. So if I were to take an arrow and draw it between these two points, this arrow would be an example of a vector. Now typically you're going to see vectors written like this where we have this V with a little arrow on top here. It's also possible that you will see notation like that. It really just depends on the textbook you use and what class you're in. But all of these are examples of the same thing which is this arrow known as a vector. Now, where the vector starts this tail end of the vector here, this is what we call the initial point. And where this vector finishes where the tip of the arrow is, this is what we call the terminal point. And I think this makes sense because naturally, if you draw this vector, you're going to draw it from initial the terminal point when you put this arrow here. Now the length of the vector is actually very important because the length tells you the magnitude of the vector. So let's just use this same velocity that we have up here. If this vector has a magnitude of six MPH, that tells you how long this vector is. Now, let's say I was Superman and I was running 50 MPH. In that case, I would need a significantly longer vector. Whereas if I was only walking one mile per hour, I would need a shorter vector. So that's the idea of a vector's magnitude. Now, the direction of a vector is dependent on the angle of the vector. So let's say that this vector that we have here has an angle of 30 degrees as it points northeast. This would be how you could find the vectors direction. So as you can see, vectors are actually pretty straightforward, all they really do is give us a direction to these quantities or magnitudes that we've learned about throughout math. Now, something else that's quite interesting about vectors is it's actually possible for vectors to be negative. Now, this might sound confusing at first because what would it really mean if I was running negative six MPH, that doesn't seem to make a ton of sense. Well, it actually turns out that all a negative vector really does. This causes the initial vector that you have to point in the opposite direction. It has the same magnitude but opposite direction. So let's say that we had negative V, if our vector V is six MPH, negative V would be negative six MPH. And what this would mean is that our vector which is initially pointing northeast is now going to point southwest. So in this instance, we would now have the initial point be at point B and the terminal point be at point A. Now, one more thing I want to cover before finishing this video is another case which is known as the zero vector, the zero vector has a magnitude of zero. And no direction. And I think that this makes sense because if I told you I was running zero MPH, well, zero MPH means I'm not moving at all. I'm just holding still. And if I'm holding still, there's no direction I'm traveling in. So it would make sense that I would have no direction and no magnitude. So this is really the main idea of vectors and how they're simply quantities that have both magnitude as well as direction. So hope you found this video helpful. Thanks for watching.

2

Problem

Problem

Determine if the following statement is true or false: Temperature is a vector.

A

True

B

False

C

Cannot be determined

3

Problem

Problem

Determine if the following statement is true or false: Acceleration is a vector.

A

True

B

False

C

Cannot be Determined

4

Problem

Problem

Determine if the following statement is true or false: The vectors $v$ and $-v$ point in the same direction.

A

True

B

False

C

Cannot be determined

5

concept

Adding Vectors Geometrically

Video duration:

5m

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Welcome back everyone. So in the last video, we got introduced to a vector and we talked about how vectors are represented as these arrows in space. Now, what we're gonna be talking about in this video is ways that we can actually add vectors together. Now, it turns out in future videos, we are going to learn that there's a way you can apply numbers to these vectors and add those numbers together. But for now, we're mostly just gonna be focusing on ways that we can visually add vectors together. So without further ado let's get right into things. Now, let's say we have vectors U and V and we wish to add them together. It turns out there is a straightforward way of doing this using something called the tip to tail method. And this method is exactly as it sounds, you take the tip of your first vector and you connect it to the tail of your second vector. So if I take these vectors and I move vector V, so it's connected tip to tail with vector U notice that we've just connected these two vectors together. And all I did was take this vector V and shift it. So it's up here. Now, once you've connected these vectors tip to tail, you want to draw a resultant vector and the resultant vector is going to go from the initial point of your first vector to the terminal point of your second vector. So if you connect the vectors in this fashion, this is going to give you the vector U plus V. So as you can see, it's actually very quick, all you do is connect the vectors tip to tail, draw this resulting vector. And that's how you can solve these types of problems. Now, let's say we instead wanted to subtract two vectors. Is there a way that we could go about doing this? Well, you might think this is significantly more complicated in addition, but it turns out it's actually not because we're told that subtracting vectors is really just the same thing as adding a negative vector. And we've already talked about what negative vectors are in the previous video. So if I wanted to take V and subtract it from vector U, I could just make vector V negative, so U minus V would be the same thing as U plus negative V and recall that a negative vector is simply going to be a vector that points in the opposite direction. So here we have vectors U and V. And if I wanted to connect these tip to tail, I'd connect this vector with U. But notice how I'm actually gonna draw the vector. So it goes in that direction as the same magnitude but opposite direction because we have negative vector V. So we have U and negative V I connect these with the initial point of the first vector to the terminal point of the second vector. And this right here would give me the vector, I'm looking for U minus V. So as you can see, it's the same idea as adding vectors together just when subtracting vectors, you have to make one of the vectors negative. Now to really make sure we're understanding this concept. I want take a look at an example and something else that's important about this example is it's actually going to tell us something interesting about the order that we do these operations for addition and subtraction. So let's start with example, a example. A asks us to find vector V plus vector U. Now what I'm gonna start by doing is drawing vector V. I can see vector V looks something like that. Now, what I need to do is I need to use the tip to tail method to find V plus U. And I can see that we have vector U right about here. So I'm going to do is connect these vectors tip to tail. And I can see that vector U is +1234567 units across and one unit up. So going here, I'll connect it to the tail of vector V. And I'm gonna go +1234567 units across and one unit up. And this right here is going to be vector U. Now, at this point, what I need to do is connect these two vectors from the initial point of the first vector to the terminal point of the second vector. And this right here is going to be the vector V plus U. Now I want you to notice something about the vector that we got. Notice that the result is actually the exact same thing that we got up here. We got vector with the exact same magnitude. They're both the same total length and they point in the same direction. So in this case, when adding vectors, whether it's V plus U or U plus V, it turns out the order actually doesn't matter for these vectors. But does this also hold true for subtraction? Well, let's go ahead and try it. Let's move on to example, B where we need to find vector V minus vector U. Now I'm going to start by drawing vector V and I can see that vector V is going to look something like this. Now, what I need to do is I need to connect this tip to tail with vector U. But notice vector U is going to be negative vector U is seven units to the right in one unit up. So what I'm going to do is reverse this direction. So we're going to go 1234567 units to the left and then we're going to go one unit down. So this would be vector negative U notice how vector negative U has the same length or magnitude but it points in the opposite direction. Now, in order to find vector V minus U, what we need to do is take the initial point of the first vector and connect it to the terminal point of the second vector. So drawing this vector is going to give me the vector V minus U and notice something about this vector. The vector that we got when we subtracted V minus U actually is different than the vector when we did U minus V. Because for vector U minus V notice that this points down into the right and the vector V minus U points up into the left, it's pointing in an entirely different direction. So it turns out the order actually does matter when we're dealing with subtraction. But the order doesn't matter when we're dealing with addition. And I think this actually makes logical sense when you just think about the numbers. Because for example, if you were to have five plus three, this would equal eight and three plus five also equals eight. The order doesn't matter when it comes to addition. But if you have five minus three, that equals two, and if you have three minus five, well, that equals negative two. Notice that it's not the same result when we subtract, but it is the same result when we add. So the same rule that applies to numbers also applies to vectors. So that is how you can add or subtract vectors and how the order does or doesn't matter depending on what operation you're doing. So hope you found this video helpful. Thanks for watching.

6

Problem

Problem

Given vectors $u ⃗$ and $v ⃗$, sketch the resultant vector $u ⃗+v ⃗$

A

B

C

D

7

Problem

Problem

Given vectors $u ⃗$ and $v ⃗$, sketch the resultant vector $u ⃗-v ⃗$.

A

B

C

D

8

example

Example 1

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2m

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In this example, we are given vectors UV and W and we're asked to sketch the resulted vector U plus V minus W. So let's see how we can do this. Now, what I'm first going to do is I'm gonna rewrite this operation over here. So we're going to have U but rather than having V minus W in the parentheses, I'm going to write this as V plus negative W, this will just allow us to see what our vectors are going to look like on this grid. Now, I'm first going to deal with what's inside the parentheses and I'm going to find negative W. Now I can see that vector W is right there and to find negative W it's going to have the same magnitude but opposite direction. So we're going to start here and we're going to finish over there because it's going to be pointing in the opposite direction as W. So this would be vector negative W. So now that we found negative W, the next thing I'm going to deal with is finding V plus negative W and I can do this using the tip to tail method. So I can see that the tip of vector V is right there. And the tail of vector negative W is right here. So if I go ahead and move negative W up there, I can connect these tip to tail. And I can see that vector negative W is +123456 units to the left. So starting here, we're gonna go +123456 units to the left. And this right here is vector negative W now to find vector V plus negative W I can just draw the resulting vector. So the resulting vector will start here, finish there. And that is going to be the vector V plus negative W which is also V minus W it's OK to write it like that as well. So this is what we end up getting now from here. What I need to do is find vector U plus V minus W. We already figured out this is vector V minus W. So what I'm going to do is use the tip to tail method on this vector. So we'll have the tip of vector U connected to the tail of vector V minus W. So I'll write the V minus W vector right there. And I see that this vector is +123 units to the left and two units down. So we'll go +123 units to the left and two units. And that's gonna be vector V minus W and I'll put this in parenthesis just to match what we have in the problem. And then all I need to do is draw another result in vector, which is going to go from the initial point of view to the final point or terminal point of V minus W. And that's going to give me the vector U plus V minus W. And that right there is the vector and the solution to this problem. So I hope you found this video helpful. Thanks for watching.

9

concept

Multiplying Vectors By Scalars

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5m

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Welcome back everyone. So in the previous video, we talked about how you could add or subtract vectors using the tip to tail method. And what we're gonna be talking about in this video is how you can multiply vectors by scalers. Now, if you're not sure what a scaler is, don't sweat it because it turns out all a scaler really is is a number that has magnitude but no direction. So an example of a scaler would be like three or five or negative 100 and you can multiply these scalers vectors to get some interesting results. And that's what we're gonna be talking about in this video. So without further ado let's get right into things. Now let's say you wanted to add three vectors together. If you wanted to do this, you could use the tip to tail method that we learned about in the previous video. So we have vector V I connect the tip of vector V with the tail of another vector V and I connect this tip to tail with a third vector V. And this right here drawing a result in vector would give us the vector V V plus V. So this is one way of solving this problem. But it turns out there's a more intuitive way that we could have added these vectors together rather than doing this three times, we could have just taken vector V and multiplied it by a SCR three. And that would, in theory give us the same result as adding the vector to itself three times. Because whenever you multiply a vector by a scr, it's either going to stretch or shrink your vector depending on what that scr is. So I can see that our scalar is three. And if we look at vector V, we can see vector V is 123 units to the right and it's one unit up. So if I were to take both of these numbers here and multiply them by three, that would give us the new vector. So three times three is nine, so we would go nine units to the right and then three times one is three. So we would go three units up. So our vector would look like this if we multiplied vector V by three. So this would be the vector three V and notice how it's the same result that we got over here, same magnitude, same direction. But it was just a bit more simple since we multiplied V by a scalar instead, now this process is actually going to become more intuitive as you do more examples of this. So to make sure that we're really understanding how to multiply vectors by scalars. Let's try some more complicated examples. So here we're told given vectors UV and W which we can see on this graph over here sketch the resulted vector based on the operations below. And we're going to start with example A which asks us to find one half of vector U. Now you can see the scalar we're dealing with is one half. So this is actually going to shrink our vector. Now I can see that our vector U is right there. So one half of vector U would just be one half of this. Now say our vector U is 1234 units to the right and 12 units up. So what I can do is shrink this by a half which would be two units to the right and one unit up. So this would be one half of vector U and that is how you can solve. Example A that's all there is to it. We found the answer. Now let's try example, B example B asks us to find negative two times vector V. Now what I can see from this example is that we have a two being multiplied by our vector. So that's going to cause our vector to be stretched. And I can see vector V is two units to the left and three units up. So what I'm going to do is scale this by two. So that's going to cause our vector to be 1234 units to the left and six units up. So we would be somewhere up there. So vector V is going to look something like this or vector two V I should say, but I want you to notice something this has a negative sign in front and recall that negative signs in front of our vector reverses the direction. So rather than pointing up in this direction, it's actually going to point down here, it's going to oppose the direction the vector V points. So this would be the vector negative to V. And that is the answer to example, B see it's pretty straightforward. Now, let's try example, C example C asks us to find one half of vector W plus vector B. Now this is a bit more complicated because we first have to find the sum of vector VNW. And then we need to multiply this by one half. We're gonna take this by the steps. Now I can find W plus V using the tip to tail method. So I see that this is W and the tip is right here. And I'm going to connect this to the tail of V. So I'm going to draw V right there. And this is going to be the vector W and vector V and then drawing the resulting vector from the initial point of W to the terminal point of V. This is going to give us vector W plus V. Now, this is not the final result because this is not ultimately what we're looking for. We're looking for one half of W plus V and to find one half of W plus B, well, we just need to cut this vector in half and reduce it. So what I can see here is that the vector W plus V is one unit down in +123456 units to the right. So half of that would be half a unit down and three units to the right. So if I draw this vector up here, we're going to go half of the unit down and we're going to go +123 units of the right. So this right here would be vector one half of W plus V. So that is the vector result right up there. And that would be the solution to example C So this would be the solution. For example, a, this would be the solution, for example B and then this would be the solution for example, C and this is how you can multiply vectors by scalers as well as do operations or deal with negative signs if necessary. So hope you found this video helpful. Thanks for watching.

10

Problem

Problem

Given vectors $u ⃗$ and $v ⃗$, sketch the resultant vector $\frac12u⃗+v⃗$.

A

B

C

D

11

Problem

Problem

Given vectors $u ⃗$ and $v ⃗$, sketch the resultant vector $-2u ⃗+3v ⃗$.

A

B

C

D

12

example

Multiplying Vectors By Scalars Example 1

Video duration:

3m

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Let's see if we can solve this example. So in this example, we are given vectors UV and W and we're asked to sketch the result in vector for negative two W plus U minus V. Now this is kind of an interesting problem because I noticed all the vectors that were given are directly horizontal or vertical. But there is some good news with this example. And that's we can use the same methods for the tip to tail method and the vector algebra we've already learned about to solve this problem as well. So let's just see how we can do this. Now, whenever we're dealing with these types of multiple vector operations like we have here, I like to start with one piece of this and then keep working on different pieces until we connect everything together and get our final vector. Now the piece I'm going to start with is this U minus V because I see this is within the parentheses. Now, I can see vector U is right about here and I can see vector V is right there. Now, vector V would this would be vector U plus V if we were to connect these tip to tail. We're looking for U minus V and to find U minus V I need to make V negative. So V currently points to the right and this positive and to make this negative, I need to have a point in the opposite direction. It'll keep the same magnitude but it will point opposite direction. So notice is now pointing to the left that is vector negative V. So this is vector negative V. What I can now do is find vector U minus V by connecting vectors U and negative V tip to tail. So we're going to have negative V where we have the tail of negative V drawn at the tip of vector U. And then I'll connect these vectors with a resulting vector which looks like this. So we're going to have the initial point of the first vector and the terminal point of the last vector and connecting those two points will give us the vector U minus V. So now we found this vector U minus V. I'm going to next focus on the vector negative two W. Now I can see that we have vector W right here. It's 1234 units to the left. And if I want to find two, W that's going to be twice the length of W. So if we go four units to the left vector two, W is going to go eight units to the left. So we're gonna go 12345678 units to the left. And that right there is going to be vector two W but we're not actually trying to find two W, we're trying to find negative two W and negative two W is going to have this vector pointing in the opposite direction. So since this is vector two W vector negative two W is going to start right here and it's going to have the same length, but it's going to point in the opposite direction. So this would be vector negative two W. Now all I need to do at this point is connect these vectors together because we found we found vector negative two W and we found vector U minus V. So to find negative two W plus U minus V, I can use the tip to tail method. So the way that I'm going to do this is I'll actually take this negative two W vector that I just found I'm going to redraw it. So it's actually up here. So it's actually going to go from this point all the way over there. That's negative two W and then we have the tip of vector negative two you drawn at the tail of vector U minus V. So all I need to do is draw this resulting vector to solve the problem. And this would give us the vector negative two W which we have right here plus U minus V which is the vector that we have right there. So this right here is the vector that we're looking for and that is the solution to this problem. So hope you found this video helpful. Thanks for watching.