Finding the a vector using angles

[exmachina]

I have three non-parallel vectors (whose components are known), v0, v1, v2 and I'd like to find a fourth unit vector, v3, given only the angles between:

psi, angle between v0 and v3
theta, angle between v1 and v3
phi, angle between v2 and v3

I know that they are related by the dot product:

v0 . v3 = |v0||v3| cos(psi)
v1 . v3 = |v1||v3| cos(theta)
v2 . v3 = |v2||v3| cos(phi)

this give me a system of equations to solve,

however, I have a term on the right side |v3|, expands into sqrt(a^2+b^2+c^2) - so its not quite linear.

If I have the constraint that I want to find a UNIT vector v3, ie. |v3|=1, then can i just set sqrt(a^2+b^2+c^2)=1, to make it a linear system?

Also - is there an easier way to solve this?

 

[exmachina]

I guess the equivalent problem would be to solve the following system of equations:

ax+by+cz=d*sqrt(x^2+y^2+z^2);
ex+fy+gz=h*sqrt(x^2+y^2+z^2);
ix+jy+kz=l*sqrt(x^2+y^2+z^2);

a,b,c,d,e,f,g,h,i,j,k,l constants
solve for x,y,z

 

[chiro]

Hey exmachina.

One thing you may have overlooked: v3 is a unit vector. What does this do to your sqrt(x^2 + y^2 + z^2) term?

 

[exmachina]

yes I realize that its equal to 1. But when I tried solving some examples setting sqrt(x^2 + y^2 + z^2)=1, the resulting values I get actually wasn't 1.

 

[blue_raver22]

I would recommend using a geometric approach to solve this problem. Since we are dealing with angles and unit vectors, we can use the concept of direction cosines.

First, we can define a unit vector v3 as (cosα, cosβ, cosγ), where α, β, and γ are the angles between v3 and the x, y, and z axes, respectively. We can then use the dot product equations given to find the direction cosines:

cos(psi) = v0 . v3 = cos α cos β cos γ
cos(theta) = v1 . v3 = cos α cos β cos γ
cos(phi) = v2 . v3 = cos α cos β cos γ

These three equations give us a system of equations with three unknowns (α, β, and γ). We can solve this system to find the direction cosines of v3.

Next, we can use the fact that a unit vector has a magnitude of 1, which means that the direction cosines must satisfy the equation:

cos²α + cos²β + cos²γ = 1

This gives us a constraint to use when solving the system of equations.

Finally, once we have found the direction cosines, we can use them to find the components of v3 by multiplying them by the magnitude of v3, which we can find using the Pythagorean theorem:

|v3| = √(cos²α + cos²β + cos²γ)

Overall, this approach allows us to find the unit vector v3 using only the angles between v3 and the other three vectors. It is a more intuitive and geometric way to solve the problem, and it avoids the non-linearity issue that arises when trying to solve the dot product equations using algebraic methods.