Compute the x and y components of the vectors A⃗, B⃗ and C⃗ shown in Figure 1.24

Name: Problem 44, Chapter 1: College Physics
Uploaded: 2019-05-30T21:15:07-08:00
Duration: 9 min 7 s
Channel: Veronica Pillar
Description: Compute the x and y components of the vectors A⃗, B⃗ and C⃗ shown in Figure 1.24

step 1 This is chapter 1, problem 44. In this problem, we are looking for the X and Y components of the

Compute the x and y components of the vectors A⃗, B⃗ and C⃗...

Key Concepts

Trigonometric Functions

Trigonometric functions, such as cosine and sine, are used to determine the magnitude of the components when a vector is resolved into x and y directions. Cosine relates the angle to the adjacent (x) component and sine to the opposite (y) component, based on the vector’s magnitude.

Coordinate Systems

A Cartesian coordinate system, consisting of mutually perpendicular x and y axes, provides the framework for vector decomposition. The coordinate system defines how the components are measured and established within the space, ensuring consistency in calculations.

Vector Decomposition

This concept involves breaking a vector into its constituent parts along specific directions, typically the horizontal (x-axis) and vertical (y-axis) components. It allows a vector’s magnitude and direction to be analyzed and utilized in various calculations, particularly in physics and engineering problems.

Direction and Sign Conventions

When decomposing a vector, the vector’s orientation relative to the coordinate axes must be considered to correctly assign positive or negative signs to its components. This is essential for accurately representing the vector’s direction in the Cartesian coordinate system.

Transcript

00:01 This is chapter 1, problem 44.

00:04 In this problem, we are looking for the x and y components of the three vectors drawn in figure 1 .24, which you should have in your textbook.

00:15 So what the figure gives us, notice for each vector, we're given a magnitude, so the length of the vector, and we're given an angle.

00:23 So when we've got magnitude and angle, or some kind of direction, we convert to x and y components using the equations on the board right now.

00:32 Now, our x component, r sub x, is equal to r cosine theta, and our y component, our y is equal to r sine theta, where r is the magnitude of the vector, and theta is the angle between the vector and the x axis, measured counterclockwise from the x axis.

00:53 So here in all cases, we are given the magnitude of the vector directly.

00:56 We can just read that off the diagram.

00:59 For the angle, we're going to have to do slightly more work, because not all of these angles are given measured counterclockwise from the x -axis.

01:09 So let's start with vector a.

01:13 It's in our first quadrant, so that's going to be maybe the easiest to think about.

01:22 We've got a -x is going to be put in our magnitude 12 meters, 12 .0 meters, but i'll just write the 12.

01:35 And cosine of our angle.

01:38 Now we're given an angle of 37 degrees with the y -axis.

01:45 The angle with the x -axis, which is the one we want, is going to be 90 degrees minus that, because from x to y is 90 degrees, and let's actually draw that over here.

02:05 So we've got our x, our y, our a vector, and so in red this whole angle is 90 degrees.

02:23 We are given in blue this 37 degrees.

02:27 So the one we want should be the whole thing 90 minus 37, or in other words, we've got 53 degrees.

02:42 And that will get you our a subx.

02:46 If you plug that into the calculator, you will find this is 7 .2 meters.

02:54 And if you plugged it into your calculator and did not get that, check whether you're in degree mode or radian mode if you're using that kind of calculator.

03:01 That's a common way people get dripped up.

03:05 All right, for a sub y, we've got our magnitude.

03:08 We already did the work to find our angle.

03:11 So we're just going to write out the same thing, 12 meters, but now we're going to do sign of 53 degrees.

03:31 And again, plug that into your calculator, and you get 9 .7 .7.

03:39 6 meters.

03:45 And we can do a quick check to see that these seem reasonable.

03:49 First, both of these are less than 12 meters, which is good.

03:51 A vector component can never be more than the vector magnitude.

03:55 And if we look on the diagram, we see that the y component, sort of sketching out in black here, looks like it's going to be greater than the x component, sketched out here.

04:07 And we find that is what we found numerically.

04:10 So that's great.

04:14 All right.

04:15 Next vector, we're going to do a similar thing, vector b in your diagram.

04:21 Here we've got our magnitude is 15 meters, so we can just plug that in for our x component.

04:39 For our angle, we are given a 40 degree angle with the x axis, but remember we want our angles to be measured counterclockwise of the x axis, and this thing is drawn clockwise from the x axis.

04:59 So basically it's like we're moving in the negative direction...

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