Question

What's the resultant of the 3 forces?​

Answer 1

Step-by-step explanation:

We need to find the x-components of each of these vectors and then add them together, then we need to find the y-components of these vectors and then add them together. Let's get to that point first. That's hard enough for step 1, dontcha think?

The x-components are found by multiplying the magnitude of the vectors by the cosine of their respective angles, while the y components are found by multiplying the magnitude of the vectors by the sine of their respective angles.

Let's do the x-components for all the vectors first, so we get the x-component of the resultant vector:

`F_(1x)=12 cos0` and

`F_(1x)=12`

`F_(2x)=9cos90` and

`F_(2x)=0`

`F_(3x)=15 cos126.87` and

`F_(3x)=-9.0` (the angle of 126.87 is found by subtracting the 53.13 from 180, since angles are to be measured from the positive axis in a counterclockwise fashion).

That means that the x-component of the resultant vector, R, is 3.0

Now for the y-components:

`F_(1y)=12sin0` and

`F_(1y)=0`

`F_(2y)=9sin90` and

`F_(2y)=9`

`F_(3y)=15sin126.87` and

`F_(3y)=12`

That means that the y-component of the resultant vector, R, is 21.

Put them together in this way to find the resultant magnitude:

`R_(mag)=√((3.0)^2+(21)^2)` which gives us

`R_(mag)=21` and now for the angle. Since both the x and y components of the resultant vector are positive, our angle will be where the x and y values are both positive in the x/y coordinate plane, which is Q1.

The angle, then:

`tan^(-1)((21)/(3.0))=82` degrees, and since we are QI, we do not add anything to this angle to maintain its accuracy.

To sum up: The resultant vector has a magnitude of 21 N at 82°