Wize University Physics Textbook (Master) > Force Vectors (Advanced Info)
Planar Angles (Shaded Triangles)
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When given a problem with 'shaded triangles' or 'planar angles', the shaded areas can by in the xy, yz, or xz plane (but not always). The shading can usually help you visualize which plane the VECTOR is in.
It is helpful to locate the RIGHT ANGLE in the shaded triangle, so you know whether to use SINE or COSINE to get the component. In the following figure, force vector F1 has TWO shaded triangles. How should we approach this?
In this case, to find the x, y, and z components, we would need to:
x: "cosine 60° down, then cosine 30°over to the positive x-axis"
y: "cosine 60° down, then sine 30° to the negative y axis" (be careful to apply the correct sign!)
z: "sine 60° to the positive z axis"
How would you approach these THREE vectors (F1, F2, F3 & F) & write them in cartesian vector form? (NOTE: F3 is a vector through 2 points which we'll cover in a few pages).
F1 = F1cos(60°)cos(30°)i - F1cos(60°)sin(30°)j + F1sin(60°)
= 0.433F1i - 0.25F1j + 0.866F1k
F2 = F2cos(135°)i + F2cos(60°)j + F2cos(60°)k
= -0.707F2i + 0.5F2j + 0.5F2k
F3: must build the unit vector first:
u3 = (4i+4j-2k)/6 = 0.667i + 0.667j - 0.333k
then F3 = F3mag u3
= 0.667F3i + 0.667F3j - 0.333F3k
F = Fxi + Fyj + Fzk
Mark Yourself Question
- Grab a piece of paper and try this problem yourself.
- When you're done, check the "I have answered this question" box below.
- View the solution and report whether you got it right or wrong.
Write each force in Cartesian Vector Form.
