Direction vectors
${\bf r}_1 = 4{\bf i} + 5{\bf j} - 3{\bf k}$
${\bf r}_2 = -6{\bf i} + 4{\bf j} - 5{\bf k}$
${\bf r}_3 = 4{\bf i} - 2{\bf j} - 3{\bf k}$
Unit vectors
${\bf \lambda}_1 = \dfrac{ 4{\bf i} + 5{\bf j} - 3{\bf k}}{\sqrt{4^2 + 5^2 + 3^2}} = 0.5657{\bf i} + 0.7071{\bf j} - 0.4243{\bf k}$
${\bf \lambda}_2 = \dfrac{-6{\bf i} + 4{\bf j} - 5{\bf k}}{\sqrt{6^2 + 4^2 + 5^2}} = -0.6838{\bf i} + 0.4558{\bf j} - 0.5698{\bf k}$
${\bf \lambda}_3 = \dfrac{4{\bf i} - 2{\bf j} - 3{\bf k}}{\sqrt{4^2 + 2^2 + 3^2}} = 0.7428{\bf i} - 0.3714{\bf j} - 0.5571{\bf k}$
Forces in rectangular form
${\bf F} = F{\bf \lambda}$
${\bf F}_1 = 200(0.5657{\bf i} + 0.7071{\bf j} - 0.4243{\bf k}) = 113.14{\bf i} + 141.42{\bf j} - 84.86{\bf k} ~ \text{lb}$
${\bf F}_2 = 400(-0.6838{\bf i} + 0.4558{\bf j} - 0.5698{\bf k}) = -273.52{\bf i} + 182.32{\bf j} - 227.92{\bf k} ~ \text{lb}$
${\bf F}_3 = 300(0.7428{\bf i} - 0.3714{\bf j} - 0.5571{\bf k}) = 222.84{\bf i} - 111.42{\bf j} - 167.13{\bf k} ~ \text{lb}$
Resultant
${\bf R} = {\bf F}_1 + {\bf F}_2 + {\bf F}_3$
Sum up the coefficients of i, j and k
${\bf R} = 62.46{\bf i} + 212.32{\bf j} - 479.91{\bf k} ~ \text{lb}$ answer
Magnitude of resultant
$R = \sqrt{62.46^2 + 212.32^2 + 479.91^2} = 528.48 ~ \text{ lb}$
Direction Cosines
$\lambda = \dfrac{{\bf R}}{R} = \dfrac{62.46{\bf i} + 212.32{\bf j} - 479.91{\bf k}}{528.48}$
$\lambda = 0.1182{\bf i} + 0.4018{\bf j} - 0.9081{\bf k}$
Calculator Operations: CASIO fx-991ES Plus
