Unit Vectors - Online Tutor, Practice Problems & Exam Prep

Unit vectors, represented by i, j, and k, simplify vector representation in terms of direction and magnitude. For example, a vector like 3i + 4j indicates movement 3 units in the x-direction and 4 units in the y-direction. This notation aids in vector addition, allowing straightforward calculations by summing components: A = 4i + 2j and B = -i + 2j results in a resultant vector of 3i + 4j. Understanding unit vectors enhances comprehension of vector operations and their applications in physics.

Unit Vectors

Hey, guys. Sometimes you're going to run into problems in which you see vectors represented using a special notation with a bunch of i's and j's, and these are called unit vectors. So in this video, I'm going to show you what these unit vectors are all about and more importantly, how they help us describe and do vector math with vectors a lot more simply. Let's check it out. So, guys, let's think of the easiest way to describe a vector. We have graphical, which is where we have already the grids and the squares. We can take this vector and break it up into its legs and just count up the boxes. Boxes. This is 3 and 4. And then we could use our equations like the Pythagorean theorem to figure out that this is 5 and the angle is 53 degrees. Now, another way we could describe the vector is by already giving the magnitude and the direction like 5 at 53 degrees. Then we'd have to just draw this vector out, so positive x and positive y and then we'd have our 5 meters here at 53 degrees. Now, if we wanted the legs, we'd have to just use our calculating or component equations or decomposition equations to figure out what the legs of the triangle are. A cosine theta, a sine theta, and we just get 34. Now the last way that you'll see vectors represented is using this weird notation with a bunch of i's and j's and k's. These are called unit vectors. And what's going on here, guys, is that these unit vectors are just special kinds of vectors that point in a direction, and they also just have a magnitude or a length of 1. So here's what's going on. If you have a vector, let's say i3+j4, all that's really going on is that I points in the plus x direction, j points in the plus y, and k points in the plus z. So anytime you see i's, j's, and k's, just think of x, y, and z's. Physicists came up with this system a long time ago. They thought it was, you know, they thought you didn't have enough confusing letters so they decided to throw a bunch of i's, j's, and k's in there. So for example, if we've got 3 I and 4 j, all it's really saying is go 3 in the I direction. So we've got 1, 2, 3. So that's 3 of them right here and then go 4 in the j or the y direction. So from here, we're gonna go 1, 2, 3, 4, and this is j j j j and you're gonna have 4 of them. That's all that's going on. So, if you wanted to construct a vector, this is just 34 and so our resultant is just gonna be from tip to tail like this and this is gonna be 5 because this is basically already giving us the legs of the triangle, 3 in the x-x and 4 in the y. So our magnitude's 5 and our angle is 53 degrees. So guys, all of these things here are just different ways to describe the exact same vector. So that's what's going on here. That's all unit vectors are. So you can think about these i's and j's as basically just already being the legs of the triangle. That's what these things are telling you.

Alright, guys. So what unit vectors are really, really helpful for is making vector addition super straightforward. Let's check it out using this example here. We've got these 2 vectors a and b. We’re going to draw them and calculate these, the resultant in unit vector form. So I've got i4+j2. So, basically, I'm going to go 1, 2, 3, 4 in the x direction. I is going to be x and then 2 in the in the j or the y direction. So 1 and then 2. So this right here gives us my a and this here is my ax and my ay. Right? It's just the components of this vector here, but I have it in terms of unit vectors. So this is just 4 I and this is just 2 j. Now, let's do the same thing for b. B is negative I. So if positive I points in the positive x direction, negative I is going to point to the left or the negative x direction. So we've got one to the left and we got 2 up. So we're going to go 1 and then 2. So this right here is going to be my BX and this is going to be my BY, and then the vector is just going to point from start to finish like this. Now, if we wanted to do vector addition, we'd have to follow all the steps. We'd have to make the table and all that stuff and you have to decompose these things and then add them together. But unit vector addition makes this stuff super straightforward. So for example, if we had this a vector, we can just describe it in terms of its I and j components or its xx and y components and I could just say that this vector is, oops, is i4+j2. And then vector b is going to be negative i+j2. So, if we wanted to find the resultant vector, which is the addition of a and b, I already have my x and my y components. And remember that when we have the x and y components, we just add them downwards. We just add them vertically in the table. This is basically already doing that for us. So my a plus b, when we could think about this, is I'm just doing ax+bx and this is going to be in the new x direction and then I'm going to do my ay plus my BY and that's going to be in the j direction or the new y direction. So for example, this new resultant vector is, if I want the resultant vector in unit vector form, I'm just going to do 4 I plus negative J or negative I or negative one I and that's going to be in the new I direction plus now I've got 2 plus 2. So 2 plus 2 in the j direction. So I've got i3+j4. And so, what that would look like, my resultant vector, is I would go 3 in the I direction and then 4 up. So my new vector would look like this. This would be my resultant vector. And this makes perfect sense because if I were to add these 2 vectors, A and B together tip to tail, I'd have to move the B vector over this way. It's one to the left and 2 up and it would get the exact same direction. So basically I would get the same exact vector. So that's just another way to describe, a vector using unit vector components.

Alright, guys. That's it for this one. Let me know if you have any questions.

A = (4.0 m)i + (3.0 m)j and B = (−13.0 m)i + (7.0 m)j. You add them together to produce another vector C.
(a) Express this new vector C in unit-vector notation.
(b) What are the magnitude and direction of C?

• A. C = (9 m) i + (10 m) j = 13.5 m @ 48° above the +x-axis
• B. C = -(9 m) i + (10 m) j = 13.5 m @ 48° above the -x-axis
• C. C = (9 m) i + (10 m) j = 13.5 m @ 0.83° above the +x-axis
• D. C = -(9 m) i + (10 m) j = 13.5 m @ 0.83° above the -x-axis

Adding 3 Vectors in Unit Vector Notation

Hey, guys. Let's check out this example together. We've got these 3 vectors, a, b, and c, and they're all written in their unit vector forms. And so we're going to calculate the magnitudes and directions of a bunch of these combinations of unit vectors. So let's check it out.

In the first part here, we've got to calculate magnitude and direction of d, which is just the addition of all the 3 unit vectors. So when we're doing unit vector addition, all we have to do is just line up the 3 vectors on top of each other. So let's write them out. My a vector is going to be 3 in the I, so it's basically 3 in the x minus 3 in the j, so it's basically 3 in the negative y direction. That's how you can think about that. My b vector is just going to be I minus 4j, and then my c vector we know is just negative 2i+5j. So if we want to calculate the resultant vector, this vector d here, by just adding all of them together, then I can just add up each of the components, basically all the parallel components, all the i's together, all the j's together, it's basically as if I already had the legs of all the triangles. I just have them in numbers. So I've got this d vector here, which is just a plus b plus c. So what I do is I just add these things straight down. So basically just add 3 and then the 1 and then the negative 2. That means that the x component of d is just going to be if I add all the components together. So 3 + 1 minus 2, that's going to be in the I direction. Plus, and now it's gonna be negative 3 minus 4 plus 5 and that's gonna be in the J. And so if I just add up all these numbers together, 3 plus 1 is 4 minus 2 is 2i and then I've got negative 3 and negative 4 is negative 7, plus 5 is negative 2. So that means I'm going to have 2i minus 2j and that is my D vector here.

But I'm not done because this is just the d vector written in unit vector form. So what I have to do is I actually have to get the magnitude and the direction of d. So let's just go ahead and sketch it out really quickly. What this would look like if you sort of like were to sketch it on a diagram, it doesn't have to be super pretty because we're just using it kind of as a sketch. We've got 2 in the x direction and then this minus sign is in the j direction. So we've got minus 2 in j. So that just means that we're going to go 2 in the x and then 2 in the y. So this means our vector is going to look something like this. This is 2 and then 2 like this. So we're going to calculate the magnitude and the direction of this vector over here. So d, and then I'll call this just theta d. So, my d is just going to be if I use the Pythagorean theorem, 2 squared plus negative 2 squared, since we've already got the legs of the triangle. And if you work this out, you're going to get 2.8. Now, the direction theta d is going to be if I do the tangent inverse and then I do the y component over the x component. But these strings are actually going to be the same. And by the way, this is technically supposed to be negative. So we've got, 2 over 2 because we're always plugging in the positive components. What you should get is you should get 45 degrees. And that makes sense because we've just got, like a perfect 45 degree angle, which means the components have to be the same. Whatever is over here is going to be over here. So that is our answer. We've got our magnitude and direction. So let's move on to part b now.

So we're going to calculate the magnitude and direction of a different combination, but it's the same principle. Here, we just have to use negative signs for when we add these vectors together. So let's see how that works. So we're going to do the same exact procedure here. So for b, we've got our a vectors. Actually, we might be able to just copy this over. So just go ahead and copy this over on your papers. Just like that. Okay. So now we're going to again, we're just going to be adding these things straight down. We're just going to be using a different formula. So instead of a plus b plus c over here, now this new vector which we're going to call e is actually going to be if we flip a and we flip b and then add it to c. So what happens when we actually make these vectors negative? Well, remember what happened when we did vector addition? We would just reverse the actual direction of those vectors? Well, all that's going to happen here for the legs of each of these triangles is that these vectors for a and b, we're going to have to reverse the sign of the components. So reverse the signs of components. So that just means that if we have instead of 3 I for our a vector in the x direction, we're going to have to use negative 3 because we have to basically just insert this little negative sign into that. So now, we got we got to flip the sign of the b vector which normally would just be 1, but now it's going to be negative 1. And then, we're going to add c. So we're just going to do nothing to c. It just stays the same. So now this is my new i direction. Plus, and now I've got to flip this over here. So if I flip this, this actually becomes positive. Then I've got to flip this one as well, so this also becomes positive. Right? Because I'm doing negative a minus b. And then I'm going to add it to 5 without doing anything. So that means that now my e vector, if I write it out, negative 3 minus 1 minus 2 is going to be negative 6 in the I, and 3 plus 4 plus 5 is actually 12. So now my I've got negative 6 I plus 12 j, And this is my e vector here. So I can do the same thing. I can basically just sketch out what this would look like. And if I have this little diagram like this, then this vector would look like Well, I have a component that's negative, so it points to the left. So that means that my x component here would look like this. And I know that this ex is equal to negative 6, and then I have my which is equal to 12. It doesn't have to be necessarily to scale, but this is what our vector would look like just to get an idea of what's actually going on. So we want to calculate now the magnitude and directions. We're going to use the same equations that we did before. So if we want to calculate the magnitude of e, we just use the Pythagorean theorem. So you just use the Pythagorean theorem of 6 squared or actually negative 6 squared, but it doesn't matter because you always get a positive number. And then 12 squared, and you get 13.4. So that is the magnitude. As for the direction, we're going to get the tangent inverse. And now, oops. So tangent inverse, and this is going to be the y component which is 12 over 6, and always going to be positive numbers, and you're going to get 63.4 degrees.

Alright, guys. That's

Here’s what students ask on this topic:

What are unit vectors and how are they represented?

Unit vectors are vectors with a magnitude of 1, used to indicate direction in a coordinate system. They are represented by the symbols i, j, and k, which correspond to the x, y, and z directions, respectively. For example, a vector $3i+4j$ indicates movement 3 units in the x-direction and 4 units in the y-direction. This notation simplifies vector operations, such as addition and subtraction, by allowing straightforward component-wise calculations.

How do you add vectors using unit vector notation?

To add vectors using unit vector notation, you sum their corresponding components. For example, if you have vectors $4i+2j$ and $-i+2j$, you add the i components and the j components separately: $4-1=3i$ and $2+2=4j$. The resultant vector is $3i+4j$.

Why are unit vectors important in physics?

Unit vectors are crucial in physics because they provide a standardized way to describe directions in space, simplifying vector operations. They allow for easy decomposition of vectors into their components, which is essential for calculations involving forces, velocities, and other vector quantities. By using unit vectors, complex vector addition, subtraction, and multiplication become more manageable, aiding in the analysis and solution of physical problems.

How do you convert a vector into unit vector form?

To convert a vector into unit vector form, you divide each component of the vector by its magnitude. For example, for a vector $3i+4j$, first calculate its magnitude: $\sqrt{3^2+4^2}=5$. Then, divide each component by 5: $\frac{3}{5}i+\frac{4}{5}j$. The unit vector form is $\frac{3}{5}i+\frac{4}{5}j$.

What is the significance of the i, j, and k notation in vector representation?

The i, j, and k notation in vector representation is significant because it provides a clear and concise way to express vectors in three-dimensional space. i represents the unit vector in the x-direction, j in the y-direction, and k in the z-direction. This notation simplifies the process of vector addition, subtraction, and scalar multiplication by allowing component-wise operations. It is widely used in physics and engineering to describe forces, velocities, and other vector quantities.