The angle and distance between the two vectors.

In a two-dimensional vector space, assume that there is one vector u(a, b) and another unknown vector v(c, d). If I knew angle and distance between these two vectors, how can I calculate the unknown vector v? I means the elements of a vector v. If I can calculate, how should I apply in Matalb??

Thank you very much.

[Honglei Chen]

could you clarify how the distance is defined between two vectors?

[YongHyun]

Both vectors have origin (0,0) and the distance means the distance between the end points of the vector. Thanks.

Answer 1

[Roger Stafford]

You have a known vector u = [a,b] and an unknown vector v = [c,d]. The distance as you have defined it is a known

   r = sqrt((c-a)^2+(d-b)^2)

and the angle in radians measured counterclockwise from u to v is a known A. You are to find v.

B = atan2(b,a);
C = cos(A+B);
S = sin(A+B);
t1 = a*C+b*S+sqrt(r^2-(a*S-b*C)^2);
t2 = a*C+b*S-sqrt(r^2-(a*S-b*C)^2);
c1 = t1*C;
d1 = t1*S;
c2 = t2*C;
d2 = t2*S;
v1 = [c1,d1];
v2 = [c2,d2];

As you can see, there will generally be two real solutions or none.

[Jan]

One solution is possible also.

[Roger Stafford]

Yes, you're right Jan. If the line of the vector happens to be exactly tangent to the circle of radius r, there will be just one solution. That's why I qualified my statement with the word 'generally'.

Answer 2

[KSSV]

Refer this:

http://math.stackexchange.com/questions/409689/how-do-i-find-a-point-a-given-distance-from-another-point-along-a-line