Introduction to Dot Product (Scalar Product): Study with Video Lessons, Practice Problems & Examples

Vector multiplication can be performed using the dot product and the cross product. The dot product, denoted as $A·B$, results in a scalar value calculated by $\mathrm{|A|}·\mathrm{|B|}·\cos \left(\theta \right)$, where $\theta$ is the angle between the vectors. This operation emphasizes the multiplication of parallel components, yielding positive, negative, or zero results based on their orientation, indicating their directional relationship.

Introduction to Dot Product (Scalar Product)

Hey, guys. So in previous videos, we saw how to multiply vectors by scalars, which are just simple numbers. Now it's pretty straightforward, but in some problems, you're going to have to multiply vectors by other vectors. And what's weird about vector multiplication is that there's a lot of not intuitive things that are going on. It's hard to visualize. But I'm going to show you that vector multiplication actually just boils down to a really simple equation. Let's check it out. There are actually 2 different ways to multiply a vector together or multiply 2 vectors together. One of them is called the dot or the scalar product and the other one's called the cross or the vector product. They get their names from their notation. So if you have 2 vectors a and b, the dot product is going to be with a big fat dot in between them and the cross product between 2 vectors a and b is going to be a big x sign like this. And so there are a couple of differences between them. The cross product is actually we're going to be covering in a later video here. So for now, we're just going to focus on the dot product. So how do we do a dot b? But before we get there, I actually want to cover something that we've previously covered before, which is how you multiply vectors by scalars. We call this multiples of vectors. For example, and it was really straightforward, if you have a vector that's 4 to the right, and you multiply it by a number 3, then you just get another longer vector out of it. So it's basically as if you're just chaining together, tip to tail, these 4 vectors, 3 times to the right, and you just get a bigger vector in this direction that's a magnitude of 12. And it points in the same direction. The dot product is different though. So when you do the dot products, you're actually going to take a vector like this 4 to the right and you're going to multiply it by another vector like 3 to the right. And what's weird about the scalar or the dot product is that you just end up with a number. You're just going to end up with 12, not the vector. So, again, if you multiply this 4 vector by this three number, you got a 12 vector. But for the dot product, you're going to multiply a 4 vector by a 3 vector and then you just get a number which is 12. You actually don't get a vector 12 in this direction. It's not what you get. It is kind of strange. And so it turns out there's actually just a really simple equation that will help us do this dot dot product. So if you have dot product a and b, what you're going to do is you're going to do the magnitude of a, which is written by the absolute value signs, times the magnitude of b, and then you're going to multiply this by the cosine of the angle where this angle is the smallest angle from a to b. So it's always this angle is drawn between a and b. Here, they're actually straight lines, but we're going to see how a bunch of different directions will affect this dot product. And so just make sure that your calculator is in degree mode. Now whenever you're doing the dot product, here's one way that I like to visualize or think about it. The dot product is basically when you're multiplying parallel components. That's how we're going to think about this. Let's just do a bunch of examples so we can see how this how this equation works out and what it means. So we're going to calculate the dot product between these two vectors right here. The first thing I want to point out is that anytime you're calculating the dot product, you're always going to line up your vectors so that they're end to end. So, basically, so that they start at the same point. Another way to think about this is tail to tail. So let's check it out. We've got these vectors a and b and we're going to calculate the dot product. So a dot b is going to be a b times the cosine of theta. So let's check it out. So we know the magnitude of a times the magnitude of b. Well, I've got these 2 vectors. I've got 3 times 4. So I've got 3 and then I've got 4 times the cosine of the angle. So I'm just going to get 12. And then what's the cosine of the angle between them? Well, this one's at 0 degrees and this one's at 0 degrees, so they're actually both completely parallel, so the cosine of 0. So these are parallel vectors and when we think of the dot product, we're going to multiply parallel components. Well, these vectors are perfectly parallel, so if you plug in cosine of 0 into your calculator, you're just going to get 1 which means that our dot product is just 12. Let's move on to Part B. Now, we've got the same exact vectors 3 and 4. Now, just one of them is raised to an angle, but let's just go through our equation. We've got AB or A dot B is going to be a b cosine theta. So when we work this out, we already know a and b is going to multiply to 12. Right? These numbers don't change. The only thing that is changing though is the cosine of the angle between them. So now what's this angle? What's the smallest angle between a and b? Well, I've got 0 This one at 0 and this one at 60, so the gap or the difference is just 60 degrees. So that's what I plug into my calculator, 60. And if you work this out, we're going to get 12 times 0.5, which means that my dot product is 6. So what's the difference here? What actually happened? Well, one way you can think about this is that this b vector here doesn't completely lie parallel to to a, but it does have a component that lies parallel to a. So this is my x component, my b x, and remember we calculate this by b cosine of theta. So one way you can think about this equation here is you're multiplying a, which is 3, times b cosine of theta. So, if you do B cosine of theta, where you plug in, so if you have this as 4 and this is 60 degrees, then my b x or my b cosine of theta term over here is gonna be 4 times the cosine of 60, which is gonna give me 2. So basically, it's like we're multiplying this 2 by this 3, not the 4 by the 3. So that's why we get a 6 out of it. So it's multiplication of parallel components. This 3 and this 2 are both parallel and that multiplies to 6. Alright, guys. Let's keep going. So now we've got part c, which is we got the same vectors. Now b just points to the left. So we're gonna start off with the same exact equation. We know a dot b is gonna be 12 times the cosine of theta. Right? So, I've got 12. Now what's the cosine of the angle between them? Well, I've got these perfectly anti parallel lines like this. So that means that the angle between them is gonna be the 2 little quadrants like this and that's just gonna be my theta is equal to a 180 degrees. So if you plug in the cosine of 180 into your calculator, you're gonna get 12 times negative one. So that means our dot product is gonna be negative 12. So what's this negative about? Well, all this really means is whenever you get a negative dot product, it just means that your components point in completely or not not completely, but it just means that they point in opposite directions. So here, the parallel components to a is actually anti parallel. It's a special case of parallel and so because they point in opposite directions, it just picks up a negative sign. That's all there is to it. Let's move on. So we've got these vectors again, 3 and 4. So let's just do go to the equation a dot b is equal to Now, you've got 12 times the cosine of theta. So we've got 12 times the cosine of what? What do we plug into our calculators? Is it the 60 degrees? Well, remember that this theta angle has to be the smallest angle between A and B. So this is actually not the angle that we're going to use. We have to find out what this angle is and this angle here is just gonna be a 180 degrees minus the 60. So our theta is actually 120. So if you were to plug in 120 into your calculator, you're gonna get 12 times negative 0.5. So now our dot product is actually negative 6. And so when we can think about this is this b vector isn't completely anti parallel to a, but it does have this component over here that does lie in the opposite direction to a. So if you work this out, this b x component, you would actually have to use this b cosine of theta. You have to use the b 60 and you would end up with a component of negative 2. And so, you're gonna multiply this negative 2 by this 3 and then that is what is what's gonna get you the negative 6. So basically, it's those 2 multiplication of anti parallel components and, you get the negative number there. Alright. So for the last one, now, we're gonna multiply this 3 and 4. Start off with the equation, a dotb is equal to 12 cosine of theta. So it's equal to 12 times the cosine of what? Well, the smallest angle here is gonna be this one right here and this actually forms a right angle. These are perfectly perpendicular like this and so this angle here is 90 degrees. If you plug in 90 degrees, the the cosine of 90 degrees, you're just gonna get 0 into your calculators. It doesn't matter how big these vectors are. It could be a billion. It could be a trillion. It doesn't matter because the cosine of this angle here is 0. So that means that your dot product is equal to 0. And so we what you can think about this is that the 0 dot product just means that the components are in completely perpendicular directions. There is no components of b that is parallel to a because they're perfectly perpendicular. And by the way, this works for any combination of vectors as long as you can figure out that the angle between them is 90. So even something like this, they don't necessarily have to be on the x and y axis like this. As long as you can figure out that this angle is 90, the dot product will always be 0. Alright, guys. That's it for this one. Let me know if you have any questions.

Using the vectors given in the figure, (a) find A ● B. (b) Find A ● C.

• A. (a) 37.4, (b) 0
• B. (a) -18.3, (b) 150
• C. (a) 37.4, (b) -134
• D. (a) 18.3, (b) 0

Here’s what students ask on this topic:

What is the dot product of two vectors?

The dot product of two vectors, also known as the scalar product, is a mathematical operation that results in a scalar value. It is calculated using the formula:

$\left|A\right|·\left|B\right|·cos\left(\theta \right)$

where $A$ and $B$ are the magnitudes of the vectors, and $\theta$ is the angle between them. The dot product emphasizes the multiplication of parallel components of the vectors.

How do you calculate the dot product of two vectors in component form?

To calculate the dot product of two vectors in component form, you multiply the corresponding components and then sum the results. For vectors $A=(A{x}_1,A{y}_1,A{z}_1)$ and $B=(B{x}_2,B{y}_2,B{z}_2)$, the dot product is:

$A·B=A{x}_1·B{x}_2+A{y}_1·B{y}_2+A{z}_1·B{z}_2$

What does a negative dot product indicate about the vectors?

A negative dot product indicates that the vectors are oriented in opposite directions. This means that the angle between the vectors is greater than 90 degrees and less than 180 degrees. In mathematical terms, if the dot product $A·B$ is negative, it implies that the cosine of the angle $\theta$ between the vectors is negative, which occurs when $90°<\theta <180°$.

How does the angle between two vectors affect their dot product?

The angle between two vectors significantly affects their dot product. If the vectors are parallel (angle = 0 degrees), the dot product is maximized and positive. If the vectors are perpendicular (angle = 90 degrees), the dot product is zero. If the vectors are anti-parallel (angle = 180 degrees), the dot product is maximized but negative. The dot product is given by:

$\left|A\right|·\left|B\right|·cos\left(\theta \right)$

where $\theta$ is the angle between the vectors. The cosine function determines the sign and magnitude of the dot product based on this angle.

What is the physical significance of the dot product in physics?

In physics, the dot product has several important applications. One key significance is in calculating work done by a force. The work $W$ done by a force $F$ over a displacement $d$ is given by the dot product:

$W=F·d=\left|F\right|·\left|d\right|·cos\left(\theta \right)$

where $\theta$ is the angle between the force and displacement vectors. This calculation helps determine how much of the force contributes to the movement in the direction of the displacement.