Force F acts perpendicular to the inclided plane. Determine the moment produced

Question

Force F acts perpendicular to the inclided plane. Determine the moment produced by F about point A. Express the result as a Cartesian Vector. In this Question why is the force position vector "a" equal to vector "rca x rcb". I understand where the position vectors of "rca" = (-4J +3k) and "rcb"=(3I -4J) come from, but i dont understand why the position vector to create a cartesian vector for F is dependent on these. I hope someone can explain this relationship.

thank you in advance.

Explanation / Answer

A vector in the cartesian plane is identified by two things.

a) Its magnitude and b) Its direction

In the cartesian system the direction is nothing but one position vector w.r.t. another. Hence, in order to specify a vector, you need to know the extending points and their position where it lies.

The reason F is dependent on these two specific vector is that the question says "F is perpendicular to the plane"

Now in order to know the direction that is perpendicular to the plane, you need to find the cross product of any two vector in that plane. (As cross product of two vector is a vector perpendicular to the plane containing them.)

Here, rca and rcb are taken. You could have taken any two among rca, rba and rbc. Result would have been same.

Now that the vector F has been calculated,

F = 249.878i +187.4085j + 249.878k

Position vector rac = Vector OC - vector OA = 3k - 4j

Moment of F about this axis = r x F = (3k-4j) x (249.878i +187.4085j + 249.878k)

M = -1561.737 i + 749.634 j + 999.512 k

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