Static Case Study for dot product

[ricky_fusion]

Homework Statement

* With an axis system oriented as shown, the position vectors of points A and B are
rA = 175i + 0j + 0k m
rB = 39i + 70j + 29k m

* When the trolley is halfway between points A and B, the forces exerted on the trolley by the cables are
F1 = -943.7i - 221.7j + 245.4k m
F2 = -919.4i - 216.0j - 328.6k m
What is the component of the force exerted on the trolley that is parallel to the line AB?

For detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=case_intro

Homework Equations

The dot product of two vectors A and B is defined as the product of the magnitudes A and B and the cosine of the angle θ between them:

A • B = |A| |B| cosθ

The dot product can also be calculated by

A • B = Ax Bx + Ay By + Az Bz
Using the dot product, the angle between two known vectors A and B, can be determined as

θ = cos^-1 AB/AB

If the direction of a line is defined by the unit vector u, then the scalar component of the vector A parallel to that line is given by

A|| = A • u

The vector component parallel to that line is given by

A|| = (A • u) u

For Detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=theory

The Attempt at a Solution

The scalar component of the total force parallel to the direction uAB can be found by using the dot product,

FT|| = FT • uAB

= -1,863 (-0.8736) - 437.7 (0.4496) - 83.2 (0.1863)

= 1,415 N

Multiply the scalar component by the unit vector uAB to determine the vector component parallel to the line AB:

FT|| = FT • uAB

= -1,236i + 636.3j + 263.7k N

= 1,415 N

For detail : https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=02.4&page=case_sol

My question
How is to find out the 1,415 N for scalar component and multiply scalar component??

Thnks,
Ricky

 

[anathema]

Dear Ricky,

Thank you for your question. The 1,415 N value for the scalar component is found by taking the dot product of the total force (FT) and the unit vector uAB. This is because the dot product is defined as the product of the magnitudes of two vectors and the cosine of the angle between them. In this case, the angle between the two vectors is 0°, meaning the cosine is equal to 1. Therefore, the dot product is equal to the product of the magnitudes of the vectors.

To find the vector component parallel to the line AB, we multiply the scalar component by the unit vector uAB. This is because the scalar component is only the magnitude of the parallel component, while the vector component also includes its direction. Multiplying the scalar component by the unit vector gives us the magnitude and direction of the parallel component of the force.

I hope this clarifies the process for you. Let me know if you have any further questions.

 

[baba89]

The 1,415 N is the result of calculating the scalar component of the total force parallel to the direction uAB using the dot product formula. To find the scalar component, you take the dot product of the total force vector (FT) and the unit vector in the direction of line AB (uAB). This gives you a numerical value, which in this case is 1,415 N.

To find the vector component parallel to the line AB, you then multiply this scalar component by the unit vector uAB. This is because the scalar component only gives you the magnitude of the force parallel to the line, but to get the full vector component, you need to multiply it by the direction vector. This will give you the vector component (1,415 N) in terms of the unit vector uAB.

I hope this helps clarify the calculation process for finding the scalar and vector components of the force parallel to the line AB. If you have any further questions, please feel free to ask.