# Thread: Vector angle in multiple planes

1. ## Vector angle in multiple planes

How would I go about solving a problem that wanted you to find the angle between two vectors of equal distance given the length of these two vectors and the sum of the two vectors (let's say #j).

You can throw out any number for distance and the sum of the two vectors. I want to see how this is done is all.

edit:
I'll throw in some numbers if that helps. These aren't my actual numbers, but maybe it'd be easier to explain.

Two vectors of equal magnitude 60.
The sum of the two vectors is 20j.
What is the angle between them?

2. Originally Posted by thedoge
How would I go about solving a problem that wanted you to find the angle between two vectors of equal distance given the length of these two vectors and the sum of the two vectors (let's say #j).

You can throw out any number for distance and the sum of the two vectors. I want to see how this is done is all.
For simplicity let us work in 2 dimensions.

You have to vectors (non-zero),
$\displaystyle \bold{u}=<u_1,u_2>$
$\displaystyle \bold{v}=<v_1,v_2>$
You are given the distance $\displaystyle s$.

Thus,
$\displaystyle u_1^2+u_2^2=s^2$ *
$\displaystyle v_1^2+v_2^2=s^2$ **

You are also given their sum,
$\displaystyle <u_1,u_2>+<v_1,v_2>=<k_1,k_2>$
Where $\displaystyle k_1,k_2$ are given.
Thus,
$\displaystyle u_1+v_1=k_1$ (1)
$\displaystyle u_2+v_2=k_2$ (2)

Square (1) and Square (2),
$\displaystyle u_1^2+2u_1v_1+v_1^2=k_1^2$
$\displaystyle u_2^2+2u_2v_2+v_2^2=k_2^2$

$\displaystyle (u_1^2+u_2^2)+2u_1v_1+2u_2v_2+(v_1^2+v_2^2)=k_1^2+ k_2^2$
From * and ** we have,
$\displaystyle 2u_1v_1+2u_2v_2+2s^2=k_1^2+k_2^2$
Divide by two,
$\displaystyle \boxed{u_1v_1+u_2v_2=\frac{k_1^2+k_2^2}{2}-s^2}$ (5)

Now, the angle between these vectors are found by,
$\displaystyle \cos \theta = \frac{|\b{u} \cdot \b{v}|}{s}$
Substitute (5) into this equation because that is the dot product.

$\displaystyle \cos \theta = \left|\frac{k_1^2+k_2^2}{2s}-s \right|$

3. Thanks. I managed to do it with trig and drawing it out, but your method helped more.