Since we already have the x and y components of vectors A and B, we just need to perform the equation for C in the two components.
x-component:
[tex] C_{x}=3 A_{x}-2 B_{x} =3(-2)-2(2)=-6-4=-10[/tex]
y-component:
[tex] C_{y}=3 A_{y}-2 B_{y} =3(4)-2(-5)=12+10=22[/tex]
We then solve for the magnitude of vector C by using the pythagorean theorem:
[tex] C^{2}= C_{x}^{2}+C_{y}^{2} [/tex]
[tex] C^{2}= (-10)^{2}+22^{2}=100+484=584[/tex]
[tex]C=24.17[/tex]
For the direction, if we perform the operation [tex] tan^{-1}(\frac{ C_{y}}{C_{x}})[/tex], we'll get the angle from the horizontal.
[tex]angle=tan^{-1}(\frac{22}{-10})=114.44deg[/tex]
ANSWER: Vector C has a magnitude of 24.17 N and a direction of 114.44 degrees from the horizontal.
