Determine the magnitudes of the components of the force F=90 lb acting parallel and perpendicula

step 1 This seems very similar to the previous problem. So first I want to start with the direction AB,

Determine the magnitudes of the components of the force F=90...

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Key Concepts

Vector Decomposition

This concept involves breaking a vector into components along chosen directions. It is a fundamental technique in physics and engineering that simplifies the analysis of vectors by considering their effects along mutually perpendicular or otherwise convenient directions.

Force Decomposition

Force decomposition is the process of resolving a force vector into components that are parallel and perpendicular to a specific line or surface. This allows for easier examination of how the force contributes to motion or equilibrium in different directions.

Dot Product and Projection

The dot product is used as a mathematical tool to project one vector onto another. By taking the dot product of a force vector with a unit vector in a desired direction, one can determine the magnitude of the component of the force acting along that direction, with the remaining component being perpendicular.

Transcript

00:01 This seems very similar to the previous problem.

00:06 So first i want to start with the direction ab, r vector for ab, is going to be, in the x direction it's going negative 1 .5 feet.

00:32 In the y direction, it's going positive 3.

00:35 And in the z direction is going positive 1.

00:48 Uab is therefore going to be negative 1 .5i plus 3j plus 1k over the square root of 1 .5 squared plus 3 squared which is 9 plus 1 squared which is 1 .2.

01:16 So that would actually be 12 .25.

01:21 Okay.

01:24 I also need u sub -f, although i'm just going to put f in its component form.

01:48 So the projection of f onto the x, y plane, is going to be f, which is 90 pounds, times the cosine of 60.

02:09 And the cosine of 60 is just one half.

02:15 So that's just going to be 45 pounds.

02:21 Now, f vector form is going to be that 45 pounds times, well, it's 45 degrees in the y direction and the x direction.

02:43 But x is going negative and y is going positive.

02:48 So it's going to be square root of 2 over 2 times 45.

02:56 And in the x direction, it's going negative.

03:01 Same thing, it's going positive in the y direction.

03:09 And in the z direction, we get 90 times the sign of 60.