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. See Example 1.

〈-7, 24〉

Accepted Answer

Welcome back everyone another video. For the given the vector calculate its magnitude and direction angle. Express your answer to the nearest sent if required. We're given our vector and specifically we know its components. We know that our vector has an X component of -16 and a Y component of 63. So let's call it vector A. If this is factor A, then AX, its X component is -16, and it's Y component A Y is equal to 63. We can easily identify the magnitude of the vector using the formula square root of AX2 plus a Y2. And if we substitute the given values. We're going to get square root of -162 + 632. Overall, we're going to get square root off. 4,225 And this is equal to 65. Specifically, according to the context of the problem, we want to use the nearest 10, even though this is an exact number, we're simply going to add 0.0, right? And now from here, we want to identify the direction angle relative to the positive X axis, so we can visualize our vector as follows. We're going to plot the XY plane. And we can notice that A X is negative and A Y is positive. So this means that our vector is in quadrant number 2. We're looking for an angle theta, and we can identify that angle by first of all, analyzing the result on right triangle, right? Because we know our AX and we know our A Y. Right, so essentially if we identify the angle alpha within the right triangle, we can simply subtract that angle from 180 degrees to end up with theta. So theta is equal to 180 degrees minus we can take the inverse of 10 to identify alpha, right? So we're going to take the inverse of 10. Of the absolute value of a Y. Divided by the absolute value of A X since we're simply looking at a real triangle, so we are ignoring those signs. And now if we continue well, we're going to get 180 degrees minus inverse of 10, let's substitute. Ay, it is 63, and then the absolute value of AX it's going to be equal to 16, right? We're going to get rid of that negative sign. And now we can continue. If we simplify the result, we're going to get approximately In this case, we're going to use an approximate time and we're going to get approximately 104.3 degrees. So we essentially have our answers. We have the magnitude of 65.0 and the angle of about 104.3 degrees. Thank you for watching.

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary. See Example 1.
〈-7, 24〉

Key Concepts

Magnitude of a Vector

Direction Angle of a Vector

Quadrants in the Coordinate Plane

Watch next