Vector v has the given direction angle and ma...

Question

Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.

θ = 27° 30' |v| = 15.4

Accepted Answer

Welcome back everyone in this problem. The magnitude and direction angle of vector U are given as 18.6 and 42 degrees respectively. We want to determine the horizontal and vertical components of the vector. For our answer choices. A says that the horizontal component is 6.9 and the vertical component is 6.2. B says it's 6.2 and 6.9 C says 13.8 and 12.4 and D says 12.4 and 13.8 respectively. Now, before we solve this problem, before we figure out these components, it would be good to try and visualize what's going on here. So what we're really saying is that we have this vector U OK. And its magnitude, its length is 18.6. So it's 18.6 units long and its direction angle, the angle that it forms with a positive axis in a counterclockwise direction is 42 degrees. Now recall that every vector has a vertical and a horizontal component. OK? And because the vertical and horizontal component are perpendicular, that means they form a right angle with each other. And since it forms a right angle triangle. We can use what we know about right triangles to help us solve any related problems. Now, let's let its vertical component be equal to Uy and its horizontal component be equal to UX. OK. Then if we think about it, OK, we can use trigonometric ratios to help us figure out the vertical and horizontal component starting with our vertical component. Uy. OK. Starting with UY. If we're going to find the value of U, which is opposite to our angle, then we can use the value of our hypotenuse the magnitude of our vector to help us. Because recall that the sine ratio tells us that the sine of an angle equals the opposite side. OK? Or is the ratio of the opposite side to the high platinum? That means that the sine of our angle is going to be equal to our opposite side, which is UI divided by our hypotenuse, which is the magnitude of U which tells us then that the vertical component UI equals the magnitude of U multiplied by the sine of our angle. OK. So it's equal the magnitude of you multiplied by the sine of our angle. Likewise. Likewise. OK. Or even before we continue, let's just solve. Let's because we have all the information we need. So let's solve here. So that means the vertical component, OK? Is going to be equal to 18.6 multiplied by the sine of our angle which is 42 degrees and 18.6 multiplied by the sine of 42 equals 12.4. So that's our vertical component. No, as I was saying before, likewise, that means then that since our horizontal component is adjacent to our angle, then our horizontal component is going to be equal to the magnitude of you multiplied by the cosine of our anger theta. OK. So that's going to be 18.6 multiplied by the cosine of 42 degrees. Which when we work that out, we get that to be equal to 13.8 units. Therefore, the horizontal component is 13.8 and the vertical component is 12.4. 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 helps.

Vector v has the given direction angle and magnitude. Find the horizontal and vertical components.
θ = 27° 30' |v| = 15.4

Key Concepts

Direction Angle in Vectors

Vector Magnitude

Resolving Vectors into Components

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