Dot Product - Online Tutor, Practice Problems & Exam Prep

The dot product of two vectors is calculated by multiplying their corresponding components and summing the results, yielding a scalar that indicates their alignment. A positive result suggests the vectors point in similar directions, while a negative result indicates they oppose each other. A zero result signifies perpendicular vectors. Alternatively, the dot product can be expressed using magnitudes and the cosine of the angle between them: $(v·u)=\mathrm{|v|}\left|u\right|cos\left(\theta \right)$. This method is useful for finding angles between vectors.

Introduction to Dot Product

Hey everyone, and welcome back. So in recent videos, we've been talking about operations you can do on vectors, and one of the operations we discussed was the product of a scalar and a vector. When multiplying a vector by a scalar, what this does is actually stretches or shrinks your vector in some way, depending on what that scalar is, but the result you get is another vector. Now what we're going to be discussing in this video is another operation called the dot product, and the dot product is actually a way to take a vector and multiply it by another vector. Now the results you get when you do a dot product are actually quite interesting. So without further ado, let's jump into some examples that you'll likely see in this course.

So let's say we have these 2 vectors, vector v and vector u, and we wish to find their dot product. Well, when doing a dot product, what you want to do is multiply the like components together, and then you want to add their results. So let's say we have these 2 vectors right here. What I can do is I can multiply the x components together and the y components together, and then add them all up. So let's go ahead and try that with the 2 vectors we have here. If I wish to find the dot product of v and u, what I can first do is multiply their x components. And their x components, I can see are 1 and 3. So we're going to have 1×3, and we're going to add this to the product of their y components, which is going to be 2×2. Now one times 3 is equal to 3. 2 times 2 is equal to 4, and 3 plus 4 is 7. So the dot product and the solution to this problem is 7.

This is how you can find a dot product. It's a pretty straightforward process, but I want you to notice something interesting. Notice when we took the product of these two vectors, we ended up getting a scalar, and that's the interesting result I was discussing before. Now, what this number tells you is this number generally represents how close the vectors are to pointing in the same direction. So a higher number would mean that the vectors are more closely aligned.

Now to make sure we really understand this concept and operation, let's actually try a few more examples. Now in this problem, we are asked to complete the following vector operations below, and we're going to start with example a. Now example a asks us to find the dot product of vectors u and w. Now I can see that vector u and w are given to us on this graph, so we can just go ahead and find their dot product. Now vector u, I can see is 3 I, and 3 I is the same thing as 3 I plus 0 j. All this 0 j tells us is that we have no direction in the y direction because this vector is completely along the x axis. And what I'm going to do is find this dot product with vector w, which is 2 I plus j. Now when doing this operation, all we need to do is multiply the like components and add the result. So we're going to have 3 times 2, which is 6, and that's going to be added to 0 times 1 which is 0. So we're going to have 6 plus 0 which is 6. And this right here is the result for our dot product. Now what exactly does this number mean? Well, it tells us how aligned vectors w and u are. And since I can see that the vectors actually are close to pointing in the same direction, we can see that they do have alignment, so we get a positive result, and that is non zero.

But now let's go ahead and try example b. Example b asked us to find vector v dotted with vector w. Now I can see that we have these two vectors right here, so let's go ahead and find their dot product. Vector v, I can see is negative 2 I plus 3 j that vector is right here, and I can dot this with vector w, which is 2 I plus j. That can multiply the like components, so we'll have negative 2 times 2, which is negative 4, and I'm going to add this to 3 times 1, which is 3. And negative 4 plus 3 is negative 1. So this right here is the result for our dot product. But what exactly does it mean when we get a negative result in the dot product? Well, what this tells me is we have negative alignment. That means the 2 vectors are not aligned at all, they're actually pulling against each other. And since it appears that these two vectors are actually pulling opposite ways, it makes sense that we got a negative result for our dot product.

But now, let's try example c. An example c is a bit more complicated because you can see that we have vector u, but this is going to be dotted with vector w plus v. Now my first step here is going to be to figure out what w plus v is. I can see vector w is 2 I plus j and I can see that vector v is negative 2 I plus 3 j. So all I need to do is add these vectors together to get this result. Now negative 2 plus 2, that's going to be 0. So we have 0 in the I direction and then I'll add this to 3j plus j which is 4j. So this is the result that we get for vector w plus vector v, and what I need to do is find this dot product which is vector u, which you can see here is 3i+0j and this is going to be dotted with vector w, which is 0i+4j. Now what I can do is multiply the like components and add the results. So we're going to have 3 times 0, which is 0, plus 0 times 1, which is 0. And 0 plus 0 is 0, so that's what we get for our dot product. Now we've talked about what a positive result means and what a negative result means, but what does a 0 result mean?

Well, a 0 means we have no alignment whatsoever. These vectors do not pull on each other, nor do they point in the same direction in any way. They're completely not aligned. And so what this means is if we have a dot product where the result is 0, the only way 2 vectors could not be aligned at all is if they were perpendicular. Now if I were to draw a vector 4 j on our graph, it would look something like this. It would point in the y direction, and notice how these two vectors are perpendicular. And because these vectors are perpendicular, that means their dot product is 0. So that's what it means to get a 0 dot product, a negative dot product, and a positive dot product. So I hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.

If vectors $v ⃗=⟨4,3⟩$ and $u ⃗=⟨9,1⟩$, calculate $v ⃗⋅u ⃗$.

• A. $39$
• B. $\langle36,3\rangle$
• C. $33$
• D. $\langle4,27\rangle$

If vectors $v⃗=12î$ and $u⃗=100ĵ$, calculate $u ⃗⋅v ⃗$.

• A. $1200k̂$
• B. $1200$
• C. $0$
• D. $100$

If vectors $a⃗=13î$, $⃗b⃗=5î-12ĵ$, and $c⃗=24ĵ$, calculate $b ⃗⋅(a ⃗-c ⃗)$.

• A. $353$
• B. $132$
• C. $65$
• D. $-223$

Find the Angle Between Vectors

Welcome back, everyone. So in the last video, we got introduced to something called the dot product, which is an operation you can do to multiply 2 vectors, where you take their like components, you multiply them, and then you add the results to get your answer. This is all review from the previous video. Now, what we're going to be talking about in this video is an alternate way of doing this dot product that involves the magnitudes of vectors, as well as the angle between them. Now this might sound like it's unnecessary since it's just some other equation to do something we've already learned, but it turns out there actually are examples that you're going to come across in this course, where you actually have to use this alternate method of finding the dot product to get a solution. So without further ado, let's just jump right into an example to see what this equation looks like, and then we'll talk about some of these special cases that you'll see in this course.

Now let's say we have 2 vectors, and we're given the magnitude of our first vector v and the magnitude of our second vector u, and we're also given the angle between them. If you wish to calculate the dot product between these two vectors, the dot product is going to be equal to the magnitude of v multiplied by the magnitude of u multiplied by the cosine of the angle between those vectors. So let's go ahead and see what we get. To find the dot product of v and u, that's going to be equal to the magnitude of v which we can see here is √5. I'll multiply that by the magnitude of u, which is √13, and then that's all going to be multiplied by the cosine of the angle between these vectors, which I can see is 29.7 degrees. Now what I'm going to do from here is simplify these 2 square roots by combining them into 1. So going to have √5 × 13 and that's all going to be multiplied by the cosine of 29.7. Now what I'm going to do from here is multiply 5 and 13, and that comes out to 65. So I have √65 × cos(29.7°). All you need to do is put this into your calculator and make sure your calculator is in degree mode when you do this. So if you plug in this value exactly as you see it, you should get that your result is about equal to 7. This is rounding our answer here, but this is what the solution should come out to. Now because we got the same solution we got over here, it turns out the 2 vectors we have, these are actually the same vectors in each case. Just in one situation we were given both vectors in component form, and then in the other situation, we were given the magnitudes and the angle. So notice how this dot product result comes out the same. But then this begs the question, why exactly would we need this equation if we're already familiar with this operation? Well, it turns out this equation allows us to find the smallest angle between 2 vectors. So if you ever want to find the smallest angle, all you need to do is rearrange this dot product formula, and that will actually allow you to find a missing angle.

So let's actually take a look at some examples that you might see in this course to make sure we know how to do this. In this example, we're told if the dot product between these two vectors, and we're given both the magnitudes of these vectors, is equal to 16, then find the angle between vectors a and b. So to solve this problem, what I can do is use this equation. So we're going to have vector a dotted with vector b is equal to the magnitude of a times the magnitude of b times the cosine of the angle between them. Now we can see here the dot product of a and b is 16. That's going to be equal to a, which we can see is 4, multiplied by b, which we can see is 8, multiplied by the cosine of the angle between them and the angle is what we don't know. Now, 4 times 8, that's equal to 32. So we're going to have 32 times the cosine of our angle is all equal to 16. And what I can do from here is divide 32 on both sides of this equation. That's gonna get the 32s to cancel there leaving me with the cosine of theta is equal to 16 over 32. Now, 16 over 32, this is a fraction that actually reduces to 1/2. Because 16 can go into 32 two times, so this reduces here. Now what I need to do from here is take the inverse cosine on both sides of this equation to get the angle by itself. That'll get the cosine to cancel leaving me with just our angle theta. And our angle theta is going to be equal to the inverse cosine of this fraction, which we said is just one-half. So all you need to do is figure out what the inverse cosine of one-half is, and you can use the unit circle, or even plug this into your calculator just make sure you're in degree mode and you should get that this angle is equal to 60 degrees. So there is a 60-degree angle between vectors a and b, and that is how you can solve for the missing angle or the smallest angle between 2 vectors. So this is what the new dot product formula is right here, and this is a situation where you would need to use this formula to find an answer. So hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.

If vectors $|a⃗|=3$ and $|b⃗|=7$, and $a⃗\cdot b⃗=14.85$, determine the angle between vectors $a ⃗$ and $b ⃗$.

• A. 21°
• B. 1.0°
• C. 45°
• D. 14.85°

If vectors $a⃗=4î$ and $b⃗=3î-2ĵ$, determine the angle between vectors $a ⃗$ and $b ⃗$.

• A. 46.10°
• B. 13.90°
• C. 43.90°
• D. 33.69°

If vectors $|v ⃗ |=12$, $|u ⃗ |=100$ and the angle between $v ⃗$ & $u ⃗$ is $\theta=\frac{\pi}{6}$, calculate $v ⃗⋅u ⃗$ .

• A. $600$
• B. $600\sqrt3$
• C. $300\sqrt3$
• D. $300$

Here’s what students ask on this topic:

What is the dot product of two vectors?

The dot product of two vectors is a scalar value obtained by multiplying their corresponding components and summing the results. For vectors $v$ and $u$ with components $v=\; (v$₁, v₂, ..., v_n) and $u=\; (u$₁, u₂, ..., u_n), the dot product is calculated as:

$(v·u)=\sum (v$_i·u_i)

This operation results in a scalar that indicates the degree of alignment between the two vectors.

How do you calculate the dot product using magnitudes and the angle between vectors?

The dot product of two vectors can also be calculated using their magnitudes and the cosine of the angle between them. For vectors $v$ and $u$ with magnitudes $\left|v\right|$ and $\left|u\right|$, and angle $\theta$ between them, the dot product is given by:

$(v·u)=\left|v\right|\left|u\right|cos\left(\theta \right)$

This method is particularly useful for finding the angle between two vectors.

What does a positive, negative, or zero dot product indicate about the vectors?

A positive dot product indicates that the vectors point in similar directions, suggesting alignment. A negative dot product means the vectors point in opposite directions, indicating they are pulling against each other. A zero dot product signifies that the vectors are perpendicular, meaning they have no alignment in any direction.

How can you find the angle between two vectors using the dot product?

To find the angle $\theta$ between two vectors $v$ and $u$, you can rearrange the dot product formula:

$(v·u)=\left|v\right|\left|u\right|cos\left(\theta \right)$

Solving for $\theta$:

$\theta =cos⁻¹\left(\frac{(v·u)}{\left|v\right|\left|u\right|}\right)$

This allows you to determine the smallest angle between the two vectors.

What is the significance of the dot product in vector analysis?

The dot product is significant in vector analysis because it provides a measure of how much one vector extends in the direction of another. It is used to determine the angle between vectors, check for orthogonality (perpendicularity), and project one vector onto another. The dot product is also essential in physics and engineering for calculating work, where force and displacement vectors are involved.