Determine the magnitude and direction of the resultant...

Determine the magnitude and direction of the resultant force, measured counterclockwise from the positive $x$ axis. Solve $l$ by first finding the resultant $\mathbf{F}^{\prime}=\mathbf{F}_{2}+\mathbf{F}_{3}$ and then forming $\mathbf{F}_{R}=\mathbf{F}^{\prime}+\mathbf{F}_{1}$

Key Concepts

Vector Addition

Vector addition involves summing multiple vectors to yield a single resultant vector. This process takes into account both the magnitude and the direction of each vector by either geometric methods or by decomposing the vectors into their components and then combining those components.

Force Decomposition

Force decomposition is the act of breaking down a force into its horizontal and vertical components, usually along the x and y axes. This is a critical step when combining forces acting at various angles, allowing for the use of algebraic methods to add the corresponding components.

Magnitude Calculation

Magnitude calculation determines the length or size of the resultant vector, typically using the Pythagorean theorem to combine the horizontal and vertical components. The magnitude represents the net effect of all forces acting on an object.

Direction Determination

Direction determination involves finding the angle at which the resultant vector acts relative to a specified reference axis. This is generally done using inverse trigonometric functions such as the arctangent, which calculates the angle based on the ratio of the vector’s vertical to horizontal components.

Transcript

00:01 All right, we've got to do it again.

00:04 I'm going to redo the drawing because it's so cluttered and it seems like there's very little relationship between doing it the first way and doing it this way.

00:17 So it is nice that we've got 90 degrees here.

00:20 You'd think that would help, although i think it really only helped for when we did it the first way.

00:29 So in theory, this way would take longer.

00:33 Except i'm going to break it into components and not put it back together until the end because it's just an extra step to put it back together.

00:48 And so i have to begin with f prime equals f2 plus f3.

01:01 But i'm not really going to find f prime.

01:03 I'm just going to get the components of f prime.

01:08 So i'm going to draw, the x and y coordinate system here and i'm going to indicate that this instead of uh writing the 150 degrees we're going to write 150 minus 90 which is 60 degrees i'm going to put 60 here um then i'll go ahead and put 90 i already put 90 in here but notice and i guess that doesn't even really matter.

01:44 It doesn't matter.

01:44 I'm just going to go with what we got there.

01:50 You know what? wait a minute.

01:52 I am going to make note of this.

01:56 This angle right here is going to be...

02:08 Well, this...

02:09 Never mind.

02:10 This angle's 30.

02:13 And so this angle, 30 plus something equals 90.

02:17 This also has to be 60 degrees.

02:21 And i'm going to erase this one because i don't think i'm going to need to know that one.

02:29 All right.