If F = {350i – 250j – 450k} and cable AB is 9 m long, determine the x, y, z coordinates of point A.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
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We will first write where point A and point B is in Cartesian vector notation.
We can use our diagram to write the location of our points are like so:
(Our i for point A is negative because point A lies on the negative x-axis side)
B:(0i+0j+0k)
We will now figure out the position vector, r_{AB} which can be found by subtracting the corresponding components of A from B. (If this is unfamiliar to you, please read this detailed guide on expressing forces in Cartesian notation)
From the question, we know the length of this rope is 9m. That means the magnitude of r_{AB}\,=\,9 m. The unit vector, denoted u, is each of r_{AB} divided by the magnitude.
We can also figure out the unit vector of F. Which is equal to:
u\,=\,0.562i-0.401j-0.723k
The key to this question is realizing that since force F is directed from point A to B, then both unit vectors must be equal. Therefore, we can write:
We can now solve for each term.
x\,=\,5.058 m
-\dfrac{y}{9}\,=\,-0.401
y\,=\,3.609 m
-\dfrac{z}{9}\,=\,-0.723
z\,=\,6.507 m
Final Answers:
y\,=\,3.609 m
z\,=\,6.507 m
This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 2, question 3-89.