Wize University Statics Textbook (Master) > Force Vectors

Planar Angles (Shaded Triangles)

Vector Addition and SubtractionNext Section

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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

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Write each force in Cartesian Vector Form.

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Wize University Statics Textbook (Master) > Force Vectors

Planar Angles (Shaded Triangles)

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