Find the magnitude and direction angle for each vector. Round ...

Question

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈15, -8〉

Accepted Answer

Welcome back everyone. For this problem, we want to calculate the magnitude and the direction angle of the vector 17 negative 13 expressing our answer to the nearest 10th. If required for our answer choices. A says it's the square root of 458 and 52.6 degrees. B says it's 458 and 52.6 degrees C, the square root of 458 and 307.4 degrees and D 458 and 307.4 degrees. Now, if we're going to calculate the magnitude and the direction angle of our vector, it helps to visualize what our vector really looks like. So let's imagine here that we have our X and Y axis. The vector 17 negative 13 means that it has an X component of 17 and A Y component of negative 13. So that means our vector would look something like this. Now, the magnitude of the vector is its length and its direction angle is the direction measured from the X axis counterclockwise. So a direction angle would look something like this. So we should be able to figure out what the both of these are. Now if you notice here that our X component and our Y component forms a right angle triangle. And for that right angle triangle, we can use those ideas to figure out its magnitude and direction angle also. Well given its X component or given our X component equals 17 and our Y component equals negative 13, then our magnitude L by the Pythagorean theorem would be equal to the square root of the sum of X squared and Y squared. Likewise theta using what we know about right angles would be equal to the inverse tangent of Y, the Y component divided by the X component. So let's use both of those here to solve for the magnitude and direction angle for our magnitude. It tells us that it is going to be equal to the square root of 17 squared plus negative 13 squared. Now, we know that 17 squared is 289 and negative 13 squared is negative 13 multiplied by negative 13, which is positive 169 and 289 plus 169 equals 458. So the magnitude of our vector is the square root of 458. Next, the direction angle the is going to be equal to the inverse tangent of 17 divided by negative 13, which we can write as negative 1713. And we put that into our calculator solving for uncertain degrees. We should get that theta is equal to 307.4 degrees. Therefore, the magnitude of our vector is the square root of 458 and its direction angle is 307.4 degrees. When we look back on our answer, choices that would have been answer choice. C thanks a lot for watching everyone. I hope this video helped.

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈15, -8〉

Key Concepts

Vector Magnitude

Direction Angle of a Vector

Quadrant Considerations in Angle Calculation

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