Answer to Question #321118 in Mechanics | Relativity for dlo

Question

Let B = 5.30 m

 at 60.0°. C

 and A

 have equal magnitudes. The direction angle of C

 is larger than that of A

 by 25.0°. Let A · B = 27.0 m²

 and B · C = 34.5 m².

 Find the magnitude (in m) and direction (in degrees) of A.

Answer 1

Let θ represent the angle between A and B. Turning by 25.0° makes the dot product larger, so the angle between C and B must be smaller.

We call it θ−25.0°. Then we have 5Acosθ=30 and  5Acos(θ−25.00)=35.

Then Acosθ=6 and A(cosθcos25.0+sinθsin25.0°)=7.

Dividing,

cos25.0°

+tanθsin25.0°=7/6

or    tanθ=(7/6−cos25.0°)/sin25.0°

=0.616.

Which gives θ=31.6°.

Then the direction angle of A is

60.0°−31.6°=28.4°.

Substituting back,

Acos31.6°=6, so  A=7.05 m at 28.4°.