Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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Video transcript

Welcome back. Everyone. In this problem. A vector is shown in the figure and we want to write it in the form A B expressing our answers as exact values are rounded to four decimal places. If necessary for our answer choices. A says that they are 8.5931 and 12.0995 B says there are 17 halves and 17 multiplied by the square root of two divided by two C says it's seven halves and five multiplied by the square of three divided by three. And the D says it's 15 halves and 15 multiplied by the square root of three divided by two. Now, in this problem, the vector that we're given tells us its magnitude and its direction vector. But we wanted to write it in the form A B. In other words, we want to write it where we can tell its vertical and horizontal components. So if I were to take this vector and put it on our diagram, what we're really looking for here is its horizontal component and its vertical component where its horizontal component is A and its vertical component is B OK. So the question is how are we going to figure that out? Well, when we look at our diagram here we notice that the horizontal and vertical components meet at 90 degrees. And thus this forms a right triangle with our vector U. Since it forms a right triangle, whatever we can do with a right triangle, we will also be able to do with our vector and its components. And we can use that idea to help us solve for its horizontal component A and vertical component B. Let's start by thinking about A. Now, in our triangle, we have a 60 degree angle and A is adjacent, the side A is adjacent to a 60 degree angle. Likewise, the side B is opposite to a 60 degree angle and or vector U is the would be the hypotenuse of our triangle that is formed of our right triangle. Now, if we're trying to find A and we know the high party news, let's think what Trigon trigonometric ratio relates the adjacent side to the high party news. Well, it would be the Cossack because recall that the cosine of an angle is the ratio of its adjacent side to its hypotenuse. So in this case, then the cosine of our angle which is 60 degrees is going to be equal to the adjacent side A divided by the hypotenuse B hypotenuse U which is 15. Therefore, a side A is going to be equal to 15 multiplied by the cosine of 60 degrees. Now, the cosine of 60 degrees is a half. So therefore, that means A equals 15, multiplied by a half or 15 halves. Likewise. So that's our value for a. Likewise, if we're solving for B, we can use the same idea. Now, if A is equal to 1560 then that must tell us B would be equal to 15 sin 60. And that's because the sine ratio links the opposite side to the hypotenuse. So if we were to actually work it out, that means the sine of 60 would be, would be B divided by 15 and thus B would be equal to, sorry, the sine of 60 would be B divided by 15 and B would be equal to 15 sine 60 as we can see here. Now, the sine of 60 is equal to the spirit of three divided by two and 15 multiplied by the square of three, divided by two equals 15, divided by two multiplied by the square of three. In other words, B is equal to 15, multiplied by the square of three divided by two. I feel like I've said that a lot in a few seconds. But essentially that means 15, divided by two would be A or the horizontal component of U and 15 multiplied by the square root of three divided by two would be B or the vertical component of U. If we look back on our answer choices that would have been answer choice. D thanks a lot for watching everyone. I hope this video helped.