The window is held open by chain AB. Determine the length of the chain, and express the 50-lb force acting at A along the chain as a Cartesian vector and determine its coordinate direction angles.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

This question involves expressing forces in Cartesian vector form. If you are unsure on how to do this, read the detailed guide on expressing forces in Cartesian notation.

Let us first figure out the locations of points A and B and write it in Cartesian form. To figure out the location of point A, we will need to use trigonometry.

Notice how we can construct a right angle triangle to figure out the location of point A. Using the diagram, the locations of points A and B are:

A:(5\cos40^0i+8j+5\sin40^0k)=(3.83i+8j+3.21k) ft

B:(0i+5j+12k) ft

We can now write our position vector from point A to B.

A position vector, denoted \mathbf{r} is a vector beginning from one point and extending to another point. It is calculated by subtracting the corresponding vector coordinates of one point from the other. If the coordinates of point A was (x_A,y_A,z_A) and the coordinates of point B was(x_B,y_B,z_B), then r_{AB}\,=\,(x_B-x_A)i+(y_B-y_A)j+(z_B-z_A)k

The magnitude of this position vector is also the length of the chain.

magnitude of r_{AB}\,=\,\sqrt{(-3.83)^2+(-3)^2+(8.79)^2}=10 ft

The magnitude is equal to the square root of the sum of the squares of the vector. If the position vector was r\,=\,ai+bj+ck, then the magnitude would be, r_{magnitude}\,=\,\sqrt{(a^2)+(b^2)+(c^2)}. In the simplest sense, you take each term of a vector, square it, add it together, and then take the square root of that value.

The unit vector is each corresponding unit of the position vector divided by the magnitude of the position vector. If the position vector was r\,=\,ai+bj+ck, then unit vector, u\,=\,\dfrac{a}{\sqrt{(a^2)+(b^2)+(c^2)}}+\dfrac{b}{\sqrt{(a^2)+(b^2)+(c^2)}}+\dfrac{c}{\sqrt{(a^2)+(b^2)+(c^2)}}

We can now express the force in Cartesian vector form:

hi, if the length AB becomes shorter, what is the effect on the coordinate direction angles?

Try it and let me know 🙂