Determine the magnitude of the projected component of the force 𝐅={400 𝐢-200 𝐣+500 𝐤} N acting a

Determine the magnitude of the projected component of the...

Key Concepts

Vector Projection

Vector projection involves determining the component of one vector in the direction of another vector. It is calculated by taking the dot product of the original vector with the unit vector in the direction of the given line or cable. This concept helps in understanding how much of the force is effective along a particular direction.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This product gives a measure of the extent to which two vectors are parallel, as it involves multiplying corresponding components and summing the result. It is essential in calculating the projection of one vector onto another.

Unit Vector and Normalization

A unit vector has a magnitude of one and indicates direction only. Normalizing a vector involves dividing the vector by its magnitude to convert it into a unit vector. This is a crucial step in vector projection because it ensures that the scaling of the projection is solely due to the original vector and not the length of the direction vector.

Force Components and Magnitude

Breaking a force vector into its components allows the analysis of its effects in different directions. In the context of a cable or any constrained path, determining the magnitude of the force component along that path (using vector projection) is essential for evaluating the effective force contributing to tension or movement along the cable.

Transcript

00:01 We have a pole and on that pole there's two cables that are connected to a wall.

00:07 And so in this drawing the little blue line is the pole and at the tip of that pole we have cables leading to points b and c on the wall and the tip of this cable is also subjected to a force, a force f.

00:20 And we have to find the magnitude of the projected component of that force acting along the cable a -b or b -a for this problem.

00:32 So to do that, we have to first find a, b, and c as points.

00:41 Okay, so to do that, and then this problem we're told various dimensions of where a, b, and c lie.

00:48 So just to speed things up, i'm going to just write them out.

00:51 Okay, so a is equal to 6i plus 0j plus 0 .0.

01:02 And since this problem is only asking for the forces along ab, we're just going to ignore c for now, because that's only relevant for a different problem.

01:12 But b is 0i, negative 1j, and 2k.

01:23 So we're going to go ahead and find the vector ab.

01:29 To do that, we just subtract the point a from the point b.

01:34 And so you get negative 6i plus j.

01:40 Plus it's minus j minus j plus plus two k so that's our our vector negative 6 i minus j into a unit vector so we first find the magnitude of ab is equal to 6 .4 using that we can find univector of a b and to do that we just divide each of these x y and z components by 6 .4 and if you do that math then the univector of a b will come out to 0 .9 375 i plus 0 .156 j okay.

02:48 And to find the projection, right, to find like what part of f is going to apply to that cable ab, we just take a dot product.

03:00 That's usually the case when we're dealing with projections.

03:04 So the dot product between the force, actually, we're actually told what the force is...

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