Can you tell me, or where can I learn, how to analytically add vectors?

Question

Thanks
I probably should have put, we're only dealing with x & y axis, unless i'm completely crazy. It's like, 8 cm @ 45 degrees plus 10 cm @ 300 degrees...plus two more.

Answer 1

We can do the x and the y separately. Just be sure to keep track of the signs.
For example 45 degrees is in the first quadrant, so x and y are both positive. But 300 degrees is in the fourth quadrant, so x is positive and y is negative.
X(resultant) = 8 cos45 + 10 cos300 = 10.657
Y(resultant) = 8 sin45 + 10sin300 = -3.003
Vector sum = (10.657, -3.003)
Resultant = √(10.657^2 + 3.003^2) = 11.072
Angle(resultant) = arctan(3.003/10.657) = 15.737degrees
Direction(resultant) = 360 - 15.737 = 344.263degrees
Or: 11.072 units at 344.263 degrees

Answer 2

I'm not sure what you mean by "analytically add vectors" but perhaps the following will help:

Suppose you have two vectors with a basis of unit vectors in the x, y and z directions: ex, ey, ez . The vectors are as follows:

u = uxex + uyey + uzez
v = vxex + vyey + vzez

The sum of u and v, u+v, is given by:

u+v = (ux + vx)ex + (uy + vy)ey+ (uz + vz)ez

In other words, you take the elements of the vector (the “x”, “y” and “z” components) and simply add the corresponding ones together and multiply by the basis vectors.