Determine the x, y, z coordinates of point A


If F = {350i – 250j – 450k} and cable AB is 9 m long, determine the x, y, z coordinates of point A.

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.

Determine the x, y, z coordinates of point A

We can use our diagram to write the location of our points are like so:

A:(-xi+yj+zk)

(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)

r_{AB}\,=\,\left\{(0-(-x))i+(0-y)j+(0-z)k)\right\}

r_{AB}\,=\,\left\{xi-yj\,-\,zk\right\}

 

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.

u\,=\,\left(\dfrac{x}{9}i\,-\,\dfrac{y}{9}j\,-\,\dfrac{z}{9}k\right)

 

We can also figure out the unit vector of F. Which is equal to:

u\,=\,\dfrac{350i-250j-450k}{\sqrt{(350)^2+(-250)^2+(-450)^2}}

 

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:

\left(\dfrac{x}{9}i\,-\,\dfrac{y}{9}j\,-\,\dfrac{z}{9}k\right)\,=\,0.562i-0.401j-0.723k

 

We can now solve for each term.

\dfrac{x}{9}\,=\,0.562

x\,=\,5.058 m

 

-\dfrac{y}{9}\,=\,-0.401

y\,=\,3.609 m

 

-\dfrac{z}{9}\,=\,-0.723

z\,=\,6.507 m

 

Final Answers:

x\,=\,5.058 m

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.

 

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