Determine the magnitude of the vectors A and B. (Assume all angles are measured counterclockwise from the positive x axis.)

Question

Two dimensional vectors, A

 and  B,

 have the same magnitude. The angle of orientation for vectors A

 and B

 are respectively  𝜃_A = 41.0°

 and  𝜃_B = 80.0°.

 The vector  C

 is related to vectors  A

 and B

 by the relationship C = A − 2B.

 The magnitude and direction for the vector C

 are |C| = 5.50

 and  𝜃_C = 287.2°.

 Determine the magnitude of the vectors A

 and  B.

 (Assume all angles are measured counterclockwise from the positive x axis.) 

=?

Answer 1

Treat the vectors as individual entities in a coordinate system, You can calculate the components of each, using trig functions, and leave them as an expression. Also, since the magnitude of B is equal to the magnitude of A, you can substitute when calculating the component magnitudes:

A_x = A•cosθ_A = A•cos 41º = .4074•A; A_y = A•sinθ_A = A•sin 41º = 0.6561•A

B_x = B•cosθ_B = A•cos 80º = 0.1736•A; B_y = B•sinθ_B = A•sin 80º = 0.9848•A

Now calculate the components for C:

C_x = C•cos θ_C = 5.50•cos 287.2º = _____

C_y = C•sin θ_C = 5.50•sin 287.2º = _____

In the problem, it states that vector C = A - 2B. This equation can be applied to the components of the vectors. Simplify the expressions:

C_x = A_x - 2B_x = 0.4074•A - (2)(0.1736•A) = _____ •A

C_y = A_y - 2B_y = 0.6561•A - (2)(0.9848•A) = _____ •A

Now substitute the values you calculated for each component of C, and solve for A. You should get the same answer for A from both the x component and the y component. And once you've found the magnitude of A, you've found the magnitude of B.