If F_B = 560 N and F_C = 700 N, determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.

#### Solution:

Show me the final answersâ†“

We will first write the locations of points A, B and C are in Cartesian vector form.

Looking at the diagram, we can write the following:

B:(2i-3j+0k)

C:(3i+2j+0k)

We can now find the position vectors from A to B and A to C.

r_{AB}\,=\,\left\{2i-3j-6k\right\} m

r_{AC}\,=\,\left\{(3-0)i+(2-0)j+(0-6)k\right\}

r_{AC}\,=\,\left\{3i+2j-6k\right\} m

We now need to find the magnitude of each position vector so that we can use it to find the unit vector in the next step.

Magnitude of r_{AB}\,=\,7 m

Magnitude of r_{AC}\,=\,\sqrt{(3^2)+(2)^2+(-6)^2}

Magnitude of r_{AC}\,=\,7 m

Let us now find the unit vector for each force. Remember, this is found by dividing each component of the position vector by it’s magnitude.

u_{AC}\,=\,\left(\dfrac{3}{7}i\,+\,\dfrac{2}{7}j\,-\,\dfrac{6}{7}k\right)

We can now write force F_B and F_C in Cartesian form.

F_B\,=\,\left\{160i\,-\,240j\,-\,480k\right\} N

F_C\,=\,700\left(\dfrac{3}{7}i\,+\,\dfrac{2}{7}j\,-\,\dfrac{6}{7}k\right)

F_C\,=\,\left\{300i\,+\,200j\,-\,600k\right\} N

(Here we multiplied each individual component by the force. In other words, expand the brackets using FOIL)

We can find the resultant force by adding both forces together like so:

F_R\,=\,\left\{160i\,-\,240j\,-\,480k\right\}\,+\,\left\{300i\,+\,200j\,-\,600k\right\}

F_R\,=\,\left\{460i-40j-1080k\right\} N

To figure out the coordinate direction angles, we need to find the magnitude of the resultant force we just found.

magnitude of F_R\,=\,1174.56 N

Coordinate direction angles can be found taking the cosine inverse of each component of the resultant divided by the magnitude:

\beta\,=\,\cos^{-1}\left(\dfrac{-40}{1174.56}\right)\,=\,92^0

\gamma\,=\,\cos^{-1}\left(\dfrac{-1080}{1174.56}\right)\,=\,157^0

#### Final Answers:

\alpha\,=\,67^0

\beta\,=\,92^0

\gamma\,=\,157^0

Thank you for the answers

how about the height of the rope AB and Ac ? how to calculate it ?

I assume you mean lengths of each of the ropes? All you need to do is write a position vector from A to B, and then find the magnitude of it. That will give you the length. The same goes for rope AC, so you write a position vector from A to C, and then find the magnitude.