Adding vectors in magnitude and direction form | Vectors | Precalculus | Khan Academy

We're told that vector A has magnitude 4 in direction 170 degrees from the positive x-axis. Vector B has magnitude 3 in direction 240 degrees from the positive x-axis. Find the magnitude and direction of vector A plus vector B. So pause this video and see if you can have a go at that.

All right, now let's work through this together. The way that I'm going to approach it, I'm going to represent each vector in component form, and then I'm going to add the corresponding components. From that, I'll try to figure out the magnitude and the direction of the sum.

So vector A, what is its x-component? Well, the change in x here, there's multiple ways that you could try to do this using trigonometry. But we've reviewed this or gone over this in other videos. The simplest way to think about it is our change in x here is going to be the length, and we know vector A has magnitude 4 times the cosine of the angle that the vector makes with the positive x-axis, cosine of 170 degrees. And so that's our x-component right over here: 4 times cosine of 170 degrees.

And then what's our y-component? Well, our y-component is going to be this change in y here, and as we've reviewed in other videos, that's going to be the length times the sine of the angle we make with the positive x-axis, sine of 170 degrees. We can maybe use a calculator in a bit to get approximations for these values.

But then we can do the exact same thing for vector B. Vector B here is going to be, by the same logic, its x-component is going to be the length of the vector, and it would be 3, they tell us that. So it's going to be 3 times the cosine of this angle, 240 degrees. And then the y-component is going to be the length of our vector, 3 times the sine of 240 degrees.

Now when we want to take the sum of the two vectors, let me write here vector A plus vector B. I can just add the corresponding components. This is going to be equal to 4 cosine of 170 degrees plus 3 cosine of 240 degrees. And then the y-component is going to be 4 sine of 170 degrees plus 3 sine of 240 degrees.

Let me get my calculator out to evaluate these. We say 170 degrees, we take the cosine times 4, that equals this. And then we're going to add to that, I'll open parentheses. We'll take the cosine of 240, 240 cosine times 3, close parentheses, is equal to this, approximately negative 5.44. So this is approximately negative 5.44.

Then if we were to take 170 degrees, take the sine of it, multiply it by 4. To that, I'm going to open parentheses. I'm going to take 240 degrees, take the sine, multiply that times 3, close my parentheses, that is going to be equal to approximately negative 1.90. So this is approximately negative 1.90.

This is consistent with our intuition. If the sum has both negative components, that means it's going to be in the third quadrant. If I were to do the head-to-tail method of adding vectors, if I were to take vector B and I were to put it right over here, we see that the resulting vector, the sum, will sit in the third quadrant. It makes sense that our x and y components would indeed be negative.

Now, the question didn't ask just to find the components of the sum; it asked to find the magnitude and the direction of the resulting sum. To do that, we just have to use a little bit more of our trigonometry and actually a little bit of our geometry. For example, our change in x is this value right over here as we go from the tail to the tip. It's negative 5.44.

If we were just thinking in terms of length right over here, the absolute value of this side would have length 5.44. Similarly, our change in y, it's negative; we're going down in y. But if we were to just think in terms of a triangle, the length on this side of a triangle is 1.90.

We can see from the Pythagorean theorem that the length of our hypotenuse, which is the same thing as the magnitude of this vector squared, is going to be equal to the sum of the squares of these two sides. Another way of thinking about it is the length of this vector, the magnitude of vector A plus vector B, is going to be equal to—or I should say approximately equal to, since we're already approximating these values—the principal root of 5.44 squared.

That's because I'm just thinking about the absolute length of the side. I could also think about a change in x, but if I had a negative 5.44 and I square that, that would still become positive. Then I'll have plus 1.90 squared. I can get our calculator out for that. This is going to be approximately equal to 5.44 squared plus 1.9 squared is equal to that. Take the square root of that, it's approximately equal to 5.76.

5.76, which is going to be our magnitude. Then, to figure out the direction, we essentially want to figure out this angle right over here. You might recognize that the tangent of this angle theta right over here should be equal to—and I'll do approximately equal to since we're using these approximations—our change in y over our change in x, so negative 1.90 over negative 5.44.

Or we could say that theta is going to be approximately equal to the inverse tangent of negative 1.90 over negative 5.44. We’re going to see in a second whether this is actually going to get us the answer that we want. So, let's try this out. If we were to take 1.90 negative, divided by 5.44 negative, that gets us that.

Which makes sense: negative divided by negative is a positive. Now let's try to take the inverse tangent of that. Here I press second, and then I'll do inverse tangent, so I'm getting 19.2 degrees approximately. This is saying that this is approximately 19.25 degrees.

My question to you is, does that seem right? Well, 19.25 degrees would put us in the first quadrant. It would give us a vector that looks something like this. This would be 19.25 degrees, but clearly that's not the vector we're talking about. We're talking about a vector in the third quadrant.

The reason why we got this result is that when you take the inverse tangent on most calculators, it's going to give you an angle that's between negative 90 degrees and positive 90 degrees. Well, here we are at an angle that puts us out in the third quadrant. So we have to adjust. To adjust, here we just have to add 180 degrees to get to the actual angle that we are talking about.

So in our situation, the magnitude here is going to be approximately 5.76, and then the direction is going to be approximately 19.25 plus 180 degrees, which is going to be 199.25 degrees. And now we are done.