How Do You Express Vector C in Terms of A, B, and Theta?

[swooshfactory]

Homework Statement

The question asks to express vector C in terms of A, B, and theta.

Homework Equations

I would guess the relevant equations to be trig equations.

The Attempt at a Solution

I found sin[(180-theta)/2] = k/B (k is a variable I set to equal the right bisected part of C when the angle c was divided in two). Also, sin[(180-theta)/2]= j/A.

c= 180 - theta. After that however, I don't know how to incorporate theta without using phi. Can you assume that a line stretching from the angle to to make a right angle with vector C bisects the angle into two equal angles? That was how I attempted to solve the problem, but I'm not sure if that works. Any help would be greatly appreciated. Thanks in advance.

 

[LowlyPion]

The question asks to express vector C in terms of A, B, and theta.

Homework Equations

I would guess the relevant equations to be trig equations.

The Attempt at a Solution

I found sin[(180-theta)/2] = k/B (k is a variable I set to equal the right bisected part of C when the angle c was divided in two). Also, sin[(180-theta)/2]= j/A.

c= 180 - theta. After that however, I don't know how to incorporate theta without using phi. Can you assume that a line stretching from the angle to to make a right angle with vector C bisects the angle into two equal angles? That was how I attempted to solve the problem, but I'm not sure if that works. Any help would be greatly appreciated. Thanks in advance.

I think you are letting your trigonometry get ahead of your vector addition.

I would suggest developing equations for the x and y components of A and B that would serve to yield C.

For simplicity I might suggest letting C lie along the x-axis. Then you know the y-components of the A and B vectors must sum to 0 and the x will sum to C.

From those equations then look to eliminate any functions of the angle ϕ and leave things in terms of θ.

 

[baba89]

To express vector C in terms of A, B, and theta, you can use the law of cosines, which states that the length of a side of a triangle can be found using the lengths of the other two sides and the cosine of the included angle. In this case, vector C can be expressed as:

C = √(A^2 + B^2 - 2ABcosθ)

Where θ is the angle between vectors A and B. This equation can be derived by drawing a triangle with sides A, B, and C, and using the law of cosines to find the length of side C.

Alternatively, you can also use the Pythagorean theorem to find the length of vector C, as it forms a right triangle with sides A and B. This method would give the same result as the law of cosines.

I am not sure how you arrived at the equations you mentioned in your solution attempt. It is important to carefully consider the given information and use the appropriate equations and concepts to solve the problem. I would suggest reviewing the concepts of trigonometry and vector operations to better understand the problem and find a correct solution.