Answer:
Length of B is 7.4833
Step-by-step explanation:
The vector sum of A and B vectors in 2D is
[tex]C=A+B=(a_1+b_1,a_2+b_2)[/tex]
And its magnitude is:
[tex]C=\sqrt{(a_1+b_1)^2+(a_2+b_2)^2} =9[/tex]
Where
[tex]a_1=Asinx[/tex]
[tex]a_2=Acosx[/tex]
[tex]b_1=Bsin(x+90)[/tex]
[tex]b_2=Bcos(x+90)[/tex]
Using the properties of the sum of two angles in the sin and cosine:
[tex]b_1=Bsin(x+90)=B(sinx*cos90+sin90*cosx)=Bcosx[/tex]
[tex]b_2=Bcos(x+90)=B(cosx*cos90-sinx*sin90)=-Bsinx[/tex]
Sustituying in the magnitud of the sum
[tex]C=\sqrt{(Asinx+Bcosx)^2+(Acosx-Bsinx)^2} =9[/tex]
[tex]C=\sqrt{A^2sin^2x+2ABsinxcosx+B^2cos^2x+A^2cos^2x-2ABsinxcosx+B^2sin^2x} =9[/tex]
[tex]C=\sqrt{A^2(sin^2x+cos^2x)+B^2(cos^2x+sin^2x)}[/tex]
[tex]C=\sqrt{A^2+B^2} =9[/tex]
Solving for B
[tex]A^2+B^2 =9^2[/tex]
[tex]B^2 =9^2-A^2[/tex]
Sustituying the value of the magnitud of A
[tex]B^2=81-5^2=81-25=56[/tex]
[tex]B= 7. 4833[/tex]
