Find the Cartesian components of force PP acting in the x, y, and z directions given P=50.0 NP=50.0 N, α=143.5∘α=143.5∘, β = 70.0∘β = 70.0∘, and γ=60.9∘γ=60.9∘. Recall that αα is the angle between the vector and the x axis, ββ is the angle between the vector and the y axis, and γγ is the angle between the vector and the z axis.

vec (F) = [ - 40.193 i + 17.101 j + 24.317 k ]

Explanation:

Given:

-The force P = 50.0 N

- Angle with x-axis α = 143.5

- Angle with y-axis β = 70.0

- Angle with z-axis γ = 60.9

Find:

Find the Cartesian components of force P acting in the x, y, and z directions,

Solution:

- The Force vector in the Cartesian coordinate system is given by the dot product of the Force P and the unit vector in its direction.

F. unit (u) = vec (F)

- Where, the unit vector is defined as:

unit (u) = [ cos (α) i + cos (β) j + cos (γ) k ]

- Using the given unit angles α, β, and γ compute unit (u):

unit (u) = [ cos (143.5) i + cos (70) j + cos (60.9) k ]

unit (u) = [ - 0.80386 i + 0.34202 j + 0.48634 k ]

- The force vector is:

vec (F) = 50.[ - 0.80386 i + 0.34202 j + 0.48634 k ]

vec (F) = [ - 40.193 i + 17.101 j + 24.317 k ]