Direction vectors
${\bf r}_1 = 2{\bf i} + 3{\bf j} + 4{\bf k}$
${\bf r}_2 = -3{\bf i} - 4{\bf j} + 5{\bf k}$
${\bf r}_3 = 0{\bf i} + 0{\bf j} - 3{\bf k}$
Unit vectors
${\bf \lambda}_1 = \dfrac{ 2{\bf i} + 3{\bf j} + 4{\bf k}}{\sqrt{2^2 + 3^2 + 4^2}} = 0.3714{\bf i} + 0.5571{\bf j} + 0.7428{\bf k}$
${\bf \lambda}_2 = \dfrac{-3{\bf i} - 4{\bf j} + 5{\bf k}}{\sqrt{3^2 + 4^2 + 5^2}} = -0.4243{\bf i} - 0.5657{\bf j} + 0.7071{\bf k}$
${\bf \lambda}_3 = \dfrac{0{\bf i} + 0{\bf j} + 4{\bf k}}{\sqrt{0^2 + 0^2 + 3^2}} = 0{\bf i} + 0{\bf j} + 1{\bf k}$
Forces in rectangular form
${\bf F} = F{\bf \lambda}$
${\bf F}_1 = 100(0.3714{\bf i} + 0.5571{\bf j} + 0.7428{\bf k}) = 37.14{\bf i} + 55.71{\bf j} + 74.28{\bf k} ~ \text{lb}$
${\bf F}_2 = 300(-0.4243{\bf i} - 0.5657{\bf j} + 0.7071{\bf k}) = -127.29{\bf i} - 169.71{\bf j} + 212.13{\bf k} ~ \text{lb}$
${\bf F}_3 = 200(0{\bf i} + 0{\bf j} + 1{\bf k}) = 0{\bf i} + 0{\bf j} + 200{\bf k} ~ \text{lb}$
Resultant
${\bf R} = {\bf F}_1 + {\bf F}_2 + {\bf F}_3$
Sum up the coefficients of i, j and k
${\bf R} = -90.15{\bf i} - 114{\bf j} + 486.41{\bf k} ~ \text{lb}$ answer
Magnitude of resultant
$R = \sqrt{90.15^2 + 114^2 + 486.41^2} = 507.66 ~ \text{ lb}$
Direction Cosines
$\lambda = \dfrac{{\bf R}}{R} = \dfrac{-90.15{\bf i} - 114{\bf j} + 486.41{\bf k}}{507.66}$
$\lambda = -0.1776{\bf i} - 0.2246{\bf j} + 0.9581{\bf k}$
Calculator Operations: CASIO fx-991ES Plus
