Vectors in Component Form - Online Tutor, Practice Problems & Exam Prep

Understanding vectors involves representing them in component form, which consists of x and y components indicating their lengths in respective directions. The magnitude of a vector can be calculated using the Pythagorean theorem, where the magnitude is the square root of the sum of the squares of its components. Additionally, vectors can be added or subtracted by combining their individual components. This foundational knowledge is crucial for further studies in mathematics and physics, enabling students to manipulate and analyze vector quantities effectively.

Position Vectors & Component Form

Welcome back, everyone. So, up to this point, we've spent a lot of time talking about vectors and what they look like. Now recall that visually, vectors are these arrows drawn in space, and we've talked about how these arrows can be stretched or shrunk, or combined with other arrows to create resultant vectors. Well, what we're going to be talking about in this video is position vectors and component form. And this might sound a bit random and confusing but don't sweat it because it turns out all we're really going to be learning about in this video is a simple way to write numbers that represent our vector. And I think you're going to find mathematically it's actually very intuitive. So without further ado, let's get right into things, because this is an important concept to understand.

Now let's say we have this vector, which we'll call vector \( \mathbf{v} \). As you can see, this is an arrow drawn in space. Now what we can do is represent this as a position vector, and a position vector is simply a vector where the initial point is drawn at the origin. So, if we want to draw vector \( \mathbf{v} \) as a position vector, I just need to relocate this so the initial point is at the origin. And that right there is the position vector, and that's all there is to it. It's just moving your vector to the origin of the graph.

Now the question becomes, how can we write this vector with some kind of numbers? Well, we can see that we have some numbers here on the graph, and we can use these numbers to figure out what our vector is. Because what we do is we represent these vectors using component form, where we have an x-component and a y-component. And all these components do is tell you the length of the vector in the x and y directions. So you can see in the x-direction, we need to go 3 units to the right, so we'd have 3. And then in the y-direction, we need to go 2 units up, so we'd have 2. So this vector is (3,2), and that's all there is to it. As you can see, it's really straightforward.

Now it turns out that there are also ways that you can represent these vectors if you don't have a position vector. So say that you were given some initial point of the vector, like a point right there, and then a terminal point over here. You could figure out what the vector is in component form by using this equation down here. And to really put this equation to use and understand it rather than just looking at it, let's actually try an example where we have to do this. So, in this example, we are told if a vector has initial (0,2,3) and a terminal (0,3,5), without drawing the vector, write the vector in component form. So we're not allowed to just graph this immediately and figure out what it looks like. What we need to do is use this equation to figure out what our vector is. But this equation is actually pretty simple to use. So all I need to do is recognize that our vector \( \mathbf{v} \) is going to be the difference in the x components, the difference in the y components.

So we can see that we have the final x minus the initial x, and then we're going to have the final y minus the initial y. Now I can see what these values are based on the points above. So if I go ahead and go to this first point, you see this first point is (0,2,3). I can see that the second point is (0,3,5). So I'm going to have for the x points is the difference between 3-2. So we're going to have 3 minus 2. And the reason that I put 3 first is that notice it's the final x minus the initial x. It's going to be 0.2 minus 0.1. So we have 3 minus 2, and then we're going to subtract the y-values. So the final y is 5, and the initial y is 3. This is what our vector is going to be. So we're going to have 3 minus 2 which is 1, and we're going to have 5 minus 3 which is 2. And that right there is the solution. That is vector \( \mathbf{v} \).

So this is how you can solve problems when you can't initially draw them or don't initially have some kind of graph of the vector. Now if we want to know what this vector looks like, we actually can use this graph just for reference. Well, I can see that our vector is (1,2). And if we draw this as a position vector, starting at the origin of our graph, we're going to go 1 to the right, and we're going to go 2 up. And that right there would be our vector \( \mathbf{v} \). So as you can see, whatever you're dealing with these types of vectors, you're going to have the x-component, which is how far we travel in the x-direction, and the y-component, which is how far we travel in the y-direction. And that's always going to be the case when using component form. So that is how you can represent vectors using numbers, and how you can draw position vectors which are at the origin of your graph. So hope you found this video helpful. Thanks for watching.

True or false: If $a ⃗=⟨3,2⟩$ and $b ⃗$ has initial point $(3,-1)$ & terminal point $(6,1)$, then $a ⃗=b ⃗$.

• A. True
• B. False
• C. Cannot be determined with given information

True or false: If $a ⃗=⟨3,-2⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$, then $a ⃗=b ⃗$.

• A. True
• B. False
• C. Cannot be determined with the given information

Position Vectors & Component Form Example 1

Let's see if we can solve this example. So in this example, we're told if vector v has initial point (4,3) and terminal point (1,2), write vector v in component form and sketch its position vector.

Now our first step is going to be to find vector v in component form, and we can do that using this equation. It's going to be x₂ minus x₁, y₂ minus y₁, and this will give us the vector we're looking for. Now the x₂ is going to be the final x or basically the terminal x and I can see the terminal point here, 1, is going to be that x value so we're going to have one minus the initial x which we can see is 4. Now next we're going to have y₂ minus y₁. Now y₂ is the y value for the terminal point which is 2, and then we're going to subtract off the y value from the initial point which is 3. So this is the vector we end up getting.

Now to solve this what we can do is we can do the subtractions here. So we're going to have 1 minus 4 which is negative 3, and we're going to have 2 minus 3 which is negative 1. So the vector is \\negative 3\<\/ms\>, \negative 1\<\/ms\>\<\/math\>, and that is the vector that we're looking for in component form.

Now we're also asked to sketch the position vector, and the position vector is just going to be this vector that starts at the origin of the graph. So let's start here at the origin. I can see we're at negative 3, so we can go 3 units to the left, and then we're going to be at negative one which is 1 unit down, and this right here would be our vector v. So notice how it goes backwards, it goes towards the negative side of the x-axis instead of the positive side because our vector is (-3, -1). So this is how you can find the vector and sketch its corresponding position vector. I hope you found this video helpful. Thanks for watching.

Finding Magnitude of a Vector

Welcome back, everyone. So in the last video, we talked about how to write vectors using component form, which ends up looking something like this where we have these brackets. Now what we're going to be talking about in this video is how you can use this component form we learned about to find the magnitude of a vector. Recall that the magnitude describes the total length of the vector itself. Finding this length has a pretty straightforward method that you can use to calculate a number to represent it. So without further ado, let's get right into an example of this, because this is an important concept to understand.

Let's say that we have this vector down here, and we want to find the magnitude. Our first step should be to figure out what this vector is in component form, based on our graph. So, I can see that we need to find the x and y components. The x component is going to be 4 units to the right, so our x component is 4, and our y component is going to be 3 units up. So our vector is 4, 3, and that is the vector in component form. But how could we use these components to figure out the length of this vector? Recall something we learned about called the Pythagorean theorem. The Pythagorean theorem is an equation that we use on right triangles, which relates all the sides of a triangle together: a2+b2=c2, where c is the hypotenuse. You can rearrange this equation to get c=a2+b2. We can use the same equation and strategy to find the magnitude of any vector. The magnitude or length of this vector can be calculated as x2+y2 where x is 4 and y is 3So, 4² is 16, and 3² is 9. 16 plus 9 is 25, and the square root of 25 is 5. So, the magnitude or length of our vector is 5 units long. This shows how you can calculate this using the Pythagorean theorem.

Now, let's actually try an example where there's not a nice graph given to us. In this example, we are asked to calculate the magnitude of vector pq if we are given the initial point or some point p, 1, 2, and the final point, 5, 3. Recall that we learned this equation in the previous video, which allows us to calculate our vector in component form if we are given 2 points. Now, our vector v is going to be equal to the final x minus the initial x, comma the final y minus the initial y. Plugging the numbers in, we have 5 minus 1, and 3 minus 2. 5 minus 1 is 4, and 3 minus 2 is 1. So our vector in component form is 4, 1. Now, the magnitude can be calculated using x2+y2 where x component is 4 and y component is 1. 4² is 16, and 1² is 1. 16 plus 1 is 17, so the magnitude of our vector is 17, which cannot be simplified any further. This is how you can find the magnitude of any vector, and you can use this equation whenever you need to calculate it. I hope you found this video helpful. Thanks for watching.

If vector $v ⃗=⟨-4,-10⟩$, calculate the magnitude $|v ⃗ |$.

• A. $4\surd29$
• B. $2\surd116$
• C. $2√29$
• D. $\surd-116$

If vector $v ⃗$ has initial point $(-1,2)$ and terminal point $(9,5)$, calculate the magnitude $|v ⃗ |$.

• A. $2√29$
• B. $\surd109$
• C. $\surd73$
• D. $\surd113$

Finding Magnitude of a Vector Example 1

Let's give this problem a try. So in this problem, we're told if vector v has initial point 1,2 and terminal point 4,4, sketch v as a position vector and calculate its magnitude. Now, to solve this problem, the first thing I'm going to do is figure out what vector v is in component form. This is always a good first step when you're dealing with these types of problems where you're given the initial and terminal point of a vector. To find this component form, what I can do is take the final x and subtract off the initial x, and then take the final y and subtract off the initial y, and that's going to give us our vector.

First off, we can see that for x₂, this is going to be the x value for the terminal point, which is 4. So what I can first do is plug in 4 to this equation. Next, I'm going to subtract off the initial x, and the initial x is 1. So we're going to have 4 minus 1 for the difference in the x's. As for the y's, I can see that we have y₂, and y₂ is going to be this y value for the terminal point, which is 4, and then this is going to be minus the initial y, and the initial y is 2. So we're going to have 4 minus 1, which is 3, and 4 minus 2, which is 2. So this right here is the vector.

Now that we have our vector in component form, what we can do is sketch the position vector. The position vector just means that our vector is going to start at the origin. So if I go here at the origin of our graph, I can see that our vector is 3. So that means on the x-axis, we're going to go over here 1, 2, 3, then I can see that our y value is 2, so that means we're going to go up 2. This is what our vector is going to look like, and that's vector v sketched as a position vector.

We've now found the component form, and we've sketched our position vector. Our last step is going to be to calculate the magnitude of this vector. To calculate the magnitude, we can use this equation: m a g n i t u d e = vx + vy where v_x is the x-component and v_y is the y-component. We have these values that we calculated already, so we're going to have v_x which is 3, so we have \(3^2\) plus \(2^2\). \(3^2\) is 9, so we're going to have the square root of 9 plus 4, which is 13. So the magnitude of our vector 'v' is equal to the square root of 13. And this is the magnitude of our vector, as well as our vector sketched as a position vector, and that is the solution to this problem. I hope you found this video helpful. Thanks for watching.

Algebraic Operations on Vectors

Welcome back, everyone. In previous videos, we've talked about how you can add or subtract vectors using the tip-to-tail method. Recall if we had one vector, we could add the tip of that vector to the tail of a second vector, and then the resultant vector that connected the two points would give you the sum of those vectors. This should be reviewed from previous videos. Now it turns out there is also a way that you can add or subtract vectors if you are given numbers as opposed to just a graph. And that's what we're going to be talking about in this video. This is a very important skill to have in this course as well as likely future math or science courses, so let's just go ahead and get right into things. If you have 2 vectors and you're given them in this component form, the way that you can add or subtract these vectors is by adding or subtracting the individual x and y components. To understand this, let's say that I have vector v, and I want to add or subtract it to vector u. What I just need to do is add or subtract the individual components. I can add or subtract the x components, so we'll have v_x ± u_x. And then, likewise, I can add or subtract the individual y components. And that's all there really is to adding or subtracting vectors. Let's go ahead and try it with this example down here. Here we have vector v = ⟨ 2, 3 ⟩, and we wish to add it to vector u = ⟨ 3, −1 ⟩. To do this, I'm first going to add the x components, 2 + 3 is 5, and then the y components, 3 + −1 is 2. So this right here is the vector v + u = ⟨ 5, 2 ⟩. That's the solution. If you use the tip-to-tail method on these vectors, you're going to find that you actually will get this result. If you have vector v = ⟨ 2, 3 ⟩ on a graph and vector u = ⟨ 3, −1 ⟩, and connect them tip to tail, you get exactly v + u. The math checks out visually as well. Another topic is multiplying a vector by a scalar, which is also crucial. Let's try an example where we have to find the vector v − 3u. First, we write out what we have. Vector v = ⟨ 8, 5 ⟩, and vector u = ⟨ 2, 4 ⟩. To find vector 3u, multiply each component of u by 3: 3*2=6 and 3*4=12, resulting in 3u = ⟨ 6, 12 ⟩. Then, subtracting 3u from v, 8 − 6 = 2 and 5 − 12 = −7. Therefore, v − 3u = ⟨ 2, −7 ⟩ is the final solution. Thanks for watching, and I hope you found this video helpful.

If vectors $v ⃗=⟨2,1⟩$, $u ⃗=⟨3,4⟩$, and $w ⃗=⟨-1,1⟩$, calculate $v ⃗+u ⃗-w ⃗$.

• A. $\langle4,4\rangle$
• B. $\langle5,4\rangle$
• C. $\langle6,6\rangle$
• D. $\langle6,4\rangle$

If vectors $v ⃗=⟨4,1⟩$, $u ⃗=⟨-8,3⟩$, and $w ⃗=⟨-2,-1⟩$, calculate $w ⃗-3(v ⃗+u ⃗)$.

• A. $\langle10,13\rangle$
• B. $\langle-14,13\rangle$
• C. $\langle-14,-13\rangle$
• D. $\langle10,-13\rangle$

Algebraic Operations on Vectors Example 1

Let's see if we can solve this example. In this example, we're told if vector a equals 31, and vector b equals negative 49, and vector c is equal to 5 times vector a minus 2 times vector b, we're asked to calculate the magnitude of vector c. Now what I can see here is that we're trying to ultimately find the magnitude of this vector, but this vector depends on the other two vectors we have in this problem. So we definitely have our work cut out for us here. But whenever solving these problems where we have multiple vector operations, I personally like to start small with the problem and then work my way to solving the whole thing. So let's actually do that.

Now the first thing I'm going to do is figure out what 5 times vector a is, because I'm really going to be focusing on this vector c here. To find 5 times a, well 5 times vector a is just going to be 5 times this vector. So we're going to have 5 times 31. Now when multiplying this scalar into each of the components, we'll have 5 times 3, which is 15, and then we'll have 5 times 1, which is 5. So this is the vector 5a.

Next, I'm going to figure out what vector 2 b is. So vector 2b is going to be 2 times vector b. I can see here that vector b is negative 49, so that's going to be our vector, and then what I need to do is distribute the scalar into this vector as well. So we're going to have 2 times negative 4, which is negative 8, and we're going to have 2 times 9, which is 18. So this is vector 2b.

And what I need to do from here is subtract these 2 vectors. To find vector c, it's going to be 5a minus 2b. We can see here that 5a is (15, 5), and 2b is (-8, 18). By subtracting these 2 vectors, we're going to have 15 minus negative 8, which is the same thing as 15 plus 8, which is 23. And then we'll have 5 minus 18, which comes out to negative 13. So this right here is vector c.

Now that I've found vector c, our last step is to just find the magnitude of c. We can use the Pythagorean theorem, and for these vectors, it's going to be the square root of the x-component of c squared plus the y-component of c squared. The x-component is 23, so we're going to have 23 squared, plus the y-component, which is negative 13, squared. So this is what we get: 232 + 132 Now, 23 squared equals 529, and negative 13 squared comes out to a positive 169. So we're going to have 529 plus 169, which comes out to 698. This value, the square root of 698, is not a radical that you can simplify further. So the magnitude of c is the square root of 698, and that is the solution to this problem.

So this is how you can handle multiple vector operations, as well as finding the magnitude of a vector once you've already done operations on that vector. I hope you found this video helpful. Thanks for watching.

Here’s what students ask on this topic:

What is the component form of a vector?

The component form of a vector is a way to represent the vector using its horizontal (x) and vertical (y) components. If a vector v has an initial point at the origin and a terminal point at (x, y), then the component form of the vector is written as v = (x, y). This notation indicates that the vector moves x units in the horizontal direction and y units in the vertical direction. For example, a vector that moves 3 units to the right and 2 units up is represented as (3, 2).

How do you find the magnitude of a vector in component form?

To find the magnitude of a vector in component form, you use the Pythagorean theorem. If a vector v has components (x, y), the magnitude |v| is calculated as:

$\sqrt{x^2+y^2}$

For example, if the vector is (4, 3), the magnitude is:

$\sqrt{42^{}+32^{}}=\sqrt{16+9}=\sqrt{25}=5$

How do you add or subtract vectors in component form?

To add or subtract vectors in component form, you simply add or subtract their corresponding components. If you have vectors v = (x₁, y₁) and u = (x₂, y₂), the sum v + u is:

(x₁ + x₂, y₁ + y₂)

Similarly, the difference v - u is:

(x₁ - x₂, y₁ - y₂)

For example, if v = (2, 3) and u = (3, -1), then:

v + u = (2 + 3, 3 - 1) = (5, 2)

v - u = (2 - 3, 3 + 1) = (-1, 4)

How do you multiply a vector by a scalar in component form?

To multiply a vector by a scalar, you multiply each component of the vector by the scalar. If vector v = (x, y) and scalar k, then the product kv is:

(k * x, k * y)

For example, if v = (2, 4) and k = 3, then:

3v = 3 * (2, 4) = (3 * 2, 3 * 4) = (6, 12)

How do you find the component form of a vector given its initial and terminal points?

To find the component form of a vector given its initial point (x₁, y₁) and terminal point (x₂, y₂), you subtract the coordinates of the initial point from the coordinates of the terminal point. The component form is:

(x₂ - x₁, y₂ - y₁)

For example, if the initial point is (2, 3) and the terminal point is (5, 7), then the component form is:

(5 - 2, 7 - 3) = (3, 4)