Geometric Vectors - Online Tutor, Practice Problems & Exam Prep

Vectors are quantities defined by both magnitude and direction, represented visually as arrows. To add vectors, the tip-to-tail method connects them, while subtraction involves adding a negative vector. Multiplying vectors by scalars alters their magnitude without changing direction, allowing for efficient calculations. Understanding these operations is crucial for future applications in physics and engineering, as they form the foundation for analyzing motion and forces. Mastery of vector operations enhances problem-solving skills and prepares students for advanced topics in mathematics and science.

Introduction to Vectors

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 6 miles per hour. And the second person tells you, I just ran 6 miles per hour 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 to 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 6 miles per hour, then that tells you how long this vector is. Now let's say I was Superman, and I was running 50 miles per hour. In that case, I would need a significantly longer vector. Whereas if I was only walking 1 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. 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 vector's 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 miles per hour? That doesn't seem to make a ton of sense. Well, it actually turns out that all a negative vector really does is 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 6 miles per hour, negative v would be negative 6 miles per hour. 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 B at point B, and the terminal point b at point a.

Now one more thing I want to cover before finishing this video is another case which is known as the 0 vector. The 0 vector has a magnitude of 0 and no direction. And I think that this makes sense. Because if I told you I was running 0 miles per hour, well, 0 miles per hour 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 I hope you found this video helpful. Thanks for watching.

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

• A. True
• B. False
• C. Cannot be determined

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

• A. True
• B. False
• C. Cannot be Determined

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

Adding Vectors Geometrically

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 going to be talking about in this video is ways that we can actually add vectors together. 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 going to 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+v. So as you can see, it's actually very quick. All you do is connect the vectors tip to tail, draw this resultant 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 than addition, but it turns out it's actually not. Because recall 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 going to draw the vector so it goes in that direction, it has the same magnitude but opposite direction because we have a negative vector v. So we have u and negative v. I can 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 to 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 going to 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+u. I can see that we have vector u right about here. So what I'm going to do is connect these vectors tip to tail, and I can see the vector u is 7 units across and 1 unit up. So going here, I'll connect it to the tail of vector v. I'm going to go 7 units across and 1 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+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 vectors 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. So 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 7 units to the right and 1 unit up. So what I'm going to do is reverse this direction. So we're going to go 7 units to the left, and then we're going to go 1 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 5 plus 3, this would equal 8. And 3+5 also equals 8. The order doesn't matter when it comes to addition. But if you have 5 minus 3, that equals 2. And if you have 3 minus 5, well, that equals negative 2. 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 could add or subtract vectors, and how the order does or doesn't matter depending on what operation you're doing. So I hope you found this video helpful. Thanks for watching.

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

• A.
• B.
• C.
• D.

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

• A.
• B.
• C.
• D.

Example 1

In this example, we are given vectors u, v, and w, and we're asked to sketch the resultant 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 going to rewrite this operation over here. So we're going to have u, but rather than having v minus w in the parenthesis, 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 if I negate 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 1, 2, 3, 4, 5, 6 units to the left. So starting here we're going to go 1, 2, 3, 4, 5, 6 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 resultant vectors. The resultant 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 okay to write it like that as well, so this is what we end up getting.

Now from here, what I need to do next, 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 1, 2, 3 units to the left and 2 units down so we'll go 1, 2, 3 units to the left and 2 units down and that's going to be vector v minus w and I'll put this in parentheses just to match what we have in the problem and then all I need to do is draw another resultant vector which is going to go from the initial point of u 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.

Multiplying Vectors By Scalars

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 going to be talking about in this video is how you can multiply vectors by scalars. Now, if you're not sure what a scalar is, don't sweat it, because it turns out all a scalar really is is a number that has magnitude but no direction. So an example of a scalar would be like 3 or 5 or negative 100, and you can multiply these scalars by vectors to get some interesting results. And that's what we're going to be talking about in this video. So without further ado, let's get right into things. Now let's say you wanted to add 3 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, and 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 resultant vector, would give us the vector v + v + 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 scalar, 3. And that would, in theory, give us the same result as adding the vector to itself 3 times. Because whenever you multiply a vector by a scalar, it's either going to stretch or shrink your vector depending on what that scalar is. So I can see that our scalar is 3. And if we look at vector v, we can see vector v is 1, 2, 3 units to the right, and it's 1 unit up. So if I were to take both of these numbers here and multiply them by 3, that would give us the new vector. So 3 times 3 is 9, so we would go 9 units to the right, and then 3 times 1 is 3, so we would go 3 units up. So our vector would look like this if we multiplied vector v by 3. So this would be the vector 3v, 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 u, v, 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 I can see the scalar we're dealing with is 1/2. 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 I'd say our vector u is 1, 2, 3, 4 units to the right and 1, 2 units up. So what I can do is shrink this by a half, which would be 2 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 negative 2 being multiplied by our vector. So that's going to cause our vector to be stretched. Now I can see vector v is 2 units to the left and 3 units up. So what I'm going to do is scale this by 2. So that's going to cause our vector to be 4 units to the left and 6 units up. So we would be somewhere up there. So vector v is going to look something like this, or vector 2v, 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 2v, 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 v. Now this is a bit more complicated because we first have to find the sum of vector v and w, and then we need to multiply this by 1/2. We're going to take this by the steps. Now I can find w + 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 resultant vector from the initial point of w to the terminal point of v. This is going to give us vector w + v. Now this is not the final result because this is not ultimately what we're looking for. We're looking for 1/2 of w + v. And if I'm 1/2 of w + v, well, we just need to cut this vector in half and reduce it. So what I can see here is that the vector w + v is 1 unit down and 1, 2, 3, 4, 5, 6 units to the right. So half of that would be half a unit down and 3 units to the right. So if I draw this vector up here, we're going to go half of a unit down and we're going to go 1, 2, 3 units to the right. So this right here would be vector one-half of w + 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 scalars, as well as do operations or deal with negative signs if necessary. So I hope you found this video helpful. Thanks for watching.

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

• A.
• B.
• C.
• D.

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

• A.
• B.
• C.
• D.

Multiplying Vectors By Scalars Example 1

Let's see if we can solve this example. So in this example, we are given vectors u, v, and w, and we're asked to sketch the result in vector for negative 2w + u - 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 1 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 - v, because I see this is within the parenthesis. Now I can see vector u is right about here, and I can see vector v is right there. Now vector v would be vector u + v if we were to connect these tip to tail. We're looking for u - v, and to find u - v, I need to make v negative. So v currently points to the right and is positive, and to make this negative, I need to have it point in the opposite direction. It'll keep the same magnitude but will point in the opposite direction. So notice it's now pointing to the left; that is vector negative v. So if this is vector negative v, what I can now do is find vector u - 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 resultant 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 - v.

So now that we found this vector u - v, I'm going to next focus on the vector negative 2w. Now I can see that we have vector w right here it's 1, 2, 3, 4 units to the left. And if I want to find 2w, that's going to be twice the length of w. So if we go 4 units to the left, vector 2w is going to go 8 units to the left. So we're gonna go 1, 2, 3, 4, 5, 6, 7, 8 units to the left, and that right there is going to be vector 2w. But we're not actually trying to find 2w, we're trying to find negative 2w, and negative 2w is going to have this vector pointing in the opposite direction. So since this is vector 2w, vector negative 2w is going to start right here, and it's going to have the same length that's going to point in the opposite direction. So this would be vector negative 2w.

Now all I need to do at this point is connect these vectors together because we found vector negative 2w, and we found vector u minus v. So to find negative 2w + u - 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 2w 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 2w. And then we have the tip of vector negative 2w drawn at the tail of vector u - v. So all I need to do is draw this resultant vector to solve the problem, and this would give us the vector negative 2w, which we have right here, plus u - 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 I hope you found this video helpful. Thanks for watching.

Here’s what students ask on this topic:

What is a vector in mathematics and how is it represented?

A vector in mathematics is a quantity that has both magnitude and direction. It is typically represented visually as an arrow. The length of the arrow indicates the vector's magnitude, while the direction of the arrow shows the vector's direction. Mathematically, vectors are often denoted by a letter with an arrow on top, such as $v^{\to }$. The starting point of the arrow is called the initial point, and the endpoint is called the terminal point. Vectors can be used to represent various physical quantities like velocity, force, and displacement.

How do you add and subtract vectors using the tip-to-tail method?

To add vectors using the tip-to-tail method, place the tail of the second vector at the tip of the first vector. The resultant vector is drawn from the initial point of the first vector to the terminal point of the second vector. For subtraction, add the negative of the vector to be subtracted. This involves reversing the direction of the vector to be subtracted and then using the tip-to-tail method. The resultant vector is drawn from the initial point of the first vector to the terminal point of the reversed vector.

What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude and no direction, such as temperature, mass, or speed. In contrast, a vector has both magnitude and direction, such as velocity, force, or displacement. Scalars are represented by simple numerical values, while vectors are represented by arrows or coordinates in a space. For example, a scalar might be 5 kg, while a vector might be 5 m/s northeast.

How do you multiply a vector by a scalar?

Multiplying a vector by a scalar changes the magnitude of the vector without altering its direction. If the scalar is positive, the vector's direction remains the same; if the scalar is negative, the vector's direction is reversed. For example, if vector $v^{\to }$ is $(3,1)$ and the scalar is 2, the resultant vector is $(6,2)$. If the scalar is -2, the resultant vector is $(-6,-2)$.

What is the zero vector and what are its properties?

The zero vector, denoted as $0^{\to }$, has a magnitude of zero and no specific direction. It is represented as a point rather than an arrow. The zero vector is unique because it is the only vector that, when added to any other vector, leaves the other vector unchanged. Mathematically, for any vector $v^{\to }$, $v^{\to }+0^{\to }=v^{\to }$. It is also the result of multiplying any vector by the scalar zero.