Establishing a reference system:
North Direction: + y South Direction -j East Direction + i West Direction -i Expressing each Force as a vector
F A ⃗ = − 4 ( j ^ ) l b F B ⃗ = 14 ( i ^ ) l b \vec{F_{A}}=-4(\hat{j})lb\\ \vec{F_{B}}=14(\hat{i})lb F A = − 4 ( j ^ ) l b F B = 14 ( i ^ ) l b
F C ⃗ = ? ? \vec{F_{C}}=?? F C = ??
Components of the resulting vector R ⃗ \vec{R} R
c o s ( 2 7 0 ) = F R F R x F R x = F R ∗ c o s ( 2 7 0 ) F R x = 9 l b ∗ c o s ( 2 7 0 ) F R x = 8 l b cos(27^{0})=\frac{ F_{R}}{F_{Rx} }\\ F_{Rx}=F_{R}*cos(27^{0})\\ F_{Rx}=9lb*cos(27^{0})\\F_{Rx}=8lb cos ( 2 7 0 ) = F R x F R F R x = F R ∗ cos ( 2 7 0 ) F R x = 9 l b ∗ cos ( 2 7 0 ) F R x = 8 l b
s i n ( 2 7 0 ) = F F R y F R y = R ∗ s i n ( 2 7 0 ) F R y = 9 l b ∗ s i n ( 2 7 0 ) F R y = 4 l b sin(27^{0})=\frac{ F}{F_{Ry }}\\ F_{Ry}=R*sin(27^{0})\\ F_{Ry}=9lb*sin(27^{0})\\F_{Ry}=4lb s in ( 2 7 0 ) = F R y F F R y = R ∗ s in ( 2 7 0 ) F R y = 9 l b ∗ s in ( 2 7 0 ) F R y = 4 l b
Finally F R ⃗ = ( 8 i ^ + 4 j ^ ) l b \vec{F_R}=(8\hat{i}+4\hat{j})lb F R = ( 8 i ^ + 4 j ^ ) l b
The sum of the vectors is F R ⃗ = F A ⃗ + F B ⃗ + F C ⃗ \vec{F_{R}}=\vec{F_{A}}+\vec{F_{B}}+\vec{F_{C}} F R = F A + F B + F C
The force Fc is
: F R ⃗ = F A ⃗ + F B ⃗ + F C ⃗ F C ⃗ = F R ⃗ − F A ⃗ − F B ⃗ \vec{F_{R}}=\vec{F_{A}}+\vec{F_{B}}+\vec{F_{C}}\\ \vec{F_{C}}=\vec{F_{R}}-\vec{F_{A}}-\vec{F_{B}} F R = F A + F B + F C F C = F R − F A − F B
Evaluating numerically
F C ⃗ = ( 8 i ^ + 4 j ^ ) l b − 4 ( j ^ ) l b − 14 ( i ^ ) l b F C ⃗ = ( 8 i ^ − 14 i ^ + 4 j ^ − 4 j ^ ) l b F C ⃗ = − 6 i ^ l b \vec{F_{C}}=(8\hat{i}+4\hat{j})lb-4(\hat{j})lb-14(\hat{i})lb\\ \vec{F_{C}}=(8\hat{i}-14\hat{i}+4\hat{j}-4\hat{j})lb \\ \vec{F_{C}}=-6\hat{i}lb F C = ( 8 i ^ + 4 j ^ ) l b − 4 ( j ^ ) l b − 14 ( i ^ ) l b F C = ( 8 i ^ − 14 i ^ + 4 j ^ − 4 j ^ ) l b F C = − 6 i ^ l b
The magnitude is
F C ⃗ = ∣ − 6 i ∣ l b F C ⃗ = 6 l b \vec{F_{C}}=|-6i|lb\\\vec{F_{C}}=6lb F C = ∣ − 6 i ∣ l b F C = 6 l b
The direction is West (-i)
The direction is determined by the unit vector of the force Fc expressed as a vector, which indicates -i that is, West
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