00:01
So then we would like to find the magnitude and direction of the result in vector r by adding the following forces using the component vector method.
00:10
And so we have a, we should draw some axes here.
00:18
So we have our y -axis and our x -axis.
00:23
And first, we have a 12 -pound force vector at 30 degrees above the x -axis.
00:30
So that is saying it is here, roughly.
00:35
12 pounds, and we're 30 degrees above the x -axis.
00:41
And then we have a 15 -pound vector at 140 degrees.
00:46
So that would imply it's because this is 0 degrees here, 90 degrees here, 180 degrees here, and 270 degrees here.
01:03
And so this is implying that it's 40 degrees above the 180 axis.
01:08
So it's kind of like that.
01:15
This is 40 and this is 15 pounds.
01:19
And then we have an 8 pound force vector at 79 degrees south of west.
01:24
So this is east.
01:27
This is north.
01:30
This is west.
01:31
This is south.
01:32
So south of west, 79 degrees south of west implies it's somewhere around here.
01:39
And it's eight pounds.
01:41
So this vector should actually be.
01:43
Be just a, it should be shorter than our 12 pound vector.
01:53
And so it's 70, so this would have right here, this angle is 79 degrees.
02:02
But we can also say that if we were to take the angle from here to here, this angle would be 259 degrees because we're adding that 79 onto the 180.
02:18
So because we're splitting up these components, these force vectors into their component parts, i recommend we make a chart, but first let's draw a little triangle to explain what's going on as we calculate all of these parts.
02:36
So this is our force vector for whichever one we care about.
02:42
And we have the y component here and the x component there.
02:46
And we have some angle theta here.
02:53
And so we know that cosine of theta is equal to adjacent over hypotenuse.
03:03
We know that sine of theta is equal to opposite over hypotenuse.
03:09
And i'm just going to label these sides.
03:11
So from angle theta, fy or the y component is going to be on the opposite side.
03:17
The x component of of f is going to be on the adjacent side and our force vector the magnitude of that this is going to be our hypotenuse so think about that when we're plugging stuff in and why we can have this general formula so we know that any x component for force is going to equal force times cosine of theta, and that f y, or the y component of force is going to equal f times sine theta.
03:52
And so let's make a little chart with the force or the magnitude of the force vector, the angle that that vector is at, and then its components.
04:13
And so let's start out by looking at the 12 pound vector, which we know has an angle of 30 degrees with respect to the positive x axis.
04:28
And what we're going to do is i'll do it for this one, but then for the others, i'm just going to skip this step because it will be the same step.
04:36
For f of x, we will have f of x is equal to the force, the magnitude of the force vector of the force vector, which we know is 12 pounds, multiplied by cosine of theta, which the theta, which the theta, for this force vector is 30.
04:55
And so we're going to get an x component value of roughly 10 .4 pounds.
05:03
And let's just write that in to our table.
05:09
For the y component, we have our magnitude of the force vector, which is 12 pounds.
05:16
And then that is multiplied by sine of 30 degrees.
05:22
And sign of 30 is just one half, and so we're going to get six pounds for our y component.
05:31
And let's just check to make sure this makes sense.
05:33
So we have a vector, and we know that it's in the quadrant with positive x values and positive y values.
05:40
Do both of our components have a positive sign? yes, we have a positive x and a positive y.
05:46
What we have done makes sense.
05:48
Now let's move on to the next force vector.
05:53
Where we have 15 pounds and 40 degrees for our theta.
06:00
I'm going to be doing the same thing for this, but i'll be replacing 12 with 15 and 30 with 40.
06:07
And so doing that, you should get negative 11 .5 pounds for your x component and 9 .6 pounds for your y component...