problem 2.7 - find the x, y, and z components of the 26 - n force shown and find its direction cosines.

Explanation:

Step1: Find the projection in x - y plane

The magnitude of the force in the x - y plane, $F_{xy}=F\times\frac{13}{13}=26$ N (since the hypotenuse of the right - triangle in the x - y plane is 13).

Step2: Calculate x - component

$F_x = F_{xy}\cos20^{\circ}\times\frac{12}{13}$ $F_x=26\times\cos20^{\circ}\times\frac{12}{13}=24\cos20^{\circ}\approx22.55$ N

Step3: Calculate y - component

$F_y = F_{xy}\sin20^{\circ}\times\frac{12}{13}$ $F_y=26\times\sin20^{\circ}\times\frac{12}{13}=24\sin20^{\circ}\approx8.21$ N

Step4: Calculate z - component

$F_z=F\times\frac{5}{13}$ $F_z = 26\times\frac{5}{13}=10$ N

Step5: Calculate direction cosines

The magnitude of the force $F = 26$ N. The direction cosine $\lambda_x=\frac{F_x}{F}=\frac{24\cos20^{\circ}}{26}\approx0.867$ $\lambda_y=\frac{F_y}{F}=\frac{24\sin20^{\circ}}{26}\approx0.316$ $\lambda_z=\frac{F_z}{F}=\frac{10}{26}\approx0.385$

Answer:

$F_x\approx22.55$ N, $F_y\approx8.21$ N, $F_z = 10$ N; $\lambda_x\approx0.867$, $\lambda_y\approx0.316$, $\lambda_z\approx0.385$