Write each vector in the form 〈a, b〉. Write answers using exac...

Question

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

Vector u with magnitude 8 at a 60-degree angle from the positive x-axis on an xy-coordinate plane.

Accepted Answer

Welcome back. Everyone. In this problem, we want to write the vector shown in the figure in the form ab expressing our answers as exact values or rounding them to four decimal places. If necessary for our answer choices. A says that it's nine multiplied by the expression, the square root of six plus the square root of two all divided by two and nine multiplied by the expression, the square root of six minus the square root of two all divided by two. B says that it's eight multiplied by the expression, the square root of C three plus the square root of two, all divided by three and eight multiplied by the expression, the square root of three minus the square root of two, all divided by three C says it's 16.7931 and 5.3982. And D says it's 15.0987 and 4.1183. Now, essentially what we are trying to do here is we're trying to write the vector with which we have been given its magnitude and direction angle in the form A B. In other words, in terms of its horizontal and the vertical components. Now, how can we relate both of those? Well, let's try to, I'm going to draw our, our, our vector here just a little bit over here. OK, along with its components. And if we think about both of these components, OK, then its horizontal component would be a while its vertical component would be B based on the form that we want to write it in. We know that it's or, or vector is 15 degrees in the counterclockwise direction from the positive X axis that its, that's its direction angle and its length or magnitude is 18 units. No, because it has horizontal and vertical components, both of them meet at a right angle. And because of that, whatever we can do with right triangles which is formed here between the components and the vector is the same thing that we can do with our vector. The same ideas will apply. So what do we know about right triangles that could first help us to find the horizontal component A we notice here that the side A is adjacent to our 15 degree angle. Likewise, the side B is opposite to our 15 degree angle. And the vector, the magnitude of the vector U 18 would be the hypotenuse of the right triangle it forms. So really what we are asking ourselves is how can we find the ADJ S side given the information that we have? That is the angle and the length of the hypotenuse. Well, recall. Ok, recall that the cosine of an anger is equal to its or is it the ratio of its adjacent side to its hypotenuse? So here that means the cosine of our anger cosine of 15 degrees is going to be equal to a or adjacent side divided by 18 or hypotenuse. That means that A is going to be equal to 18 multiplied by the cosine of 15 degrees and 18 multiplied by the cosine of 15 degrees. In its exact value would have been nine multiplied by the square of six plus the square root of two all divided by two. That's what it would have been in its exact value. Now, what does that mean for us? Well, since that's the value of A, we can also use the same idea to figure out the value of B because here, for the value of B, since B is the opposite side, we can use the same idea because the sine of 15 is going to be equal to B divided by 18. And thus B must be then equal to 18, multiplied by the sine of 15 degrees. Now, when we put that into our, into our calculator, and we choose to get its exact value, we'll find that its exact value is nine multiplied by the square root of six minus the square root of two all divided by two. Therefore, our coordinates or, or uh or, or angle rather not our coordinates but our vector in the form A B would have been nine multiplied by the expression, the square of six plus the square of two all divided by two and the nine multiplied by the square root of six minus the square root of two all divided by two. If we look back on our answer choices, that would have been answer choice. A thanks a lot for watching everyone. I hope this video helped.

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

Key Concepts

Vector Representation in Component Form

Trigonometric Functions for Component Calculation

Exact Values and Decimal Approximations

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