## Homework Statement

Three units vectors a, b, and c have property that the angle between any two is a fixed angle $$\theta$$

(i) find in terms of $$\theta$$ the length of the vector v = a + b + c
(ii) find the largest possible value of $$\theta$$
(iii) find the cosine of the angle $$\beta$$ between a and v

## Homework Equations

unit vector = vector with length 1unit

magnitude of vector = $$\sqrt{x^2+y^2+z^2}$$

$$\cos \theta = \frac{r_1\cdot r_2}{|r_1||r_2|}$$

## The Attempt at a Solution

(i) I think I get it right. The answer is $$\sqrt{3+6\cos \theta}$$

(ii) I don't know how to do this. I think $$\theta < 90^o$$ , but I can't find the exact value

(iii)
$$\cos \beta = \frac{a\cdot v}{|a||v|}$$

After some calculation,

$$\cos \beta = \frac{2+\cos \theta}{\sqrt{3+6\cos \theta}}$$

Can it be simplified further?

Thanks a lot

lanedance
Homework Helper

i) looks ok

ii) think about the case when they are all in the same plane...

iii) shouldn't this be 1 + 2cos(theta) on the numerator?

Office_Shredder
Staff Emeritus
Gold Member

Look at (i), and ask yourself for what values of theta can that length even exist? You know that v=a+b+c must be an actual vector, which means it must have an actual length

lanedance
Homework Helper

though v can be the zero vector, with zero length

Hi lanedance and Office_Shredder

Ah yes or (iii) it should be 1 + 2cos(theta). I found it but dunno why I wrote 2 + cos(theta) here....

For (ii) , The length of v can exist if :

$$3+6 \cos \theta \geq 0$$

I found the value for $$\theta$$ = [0o, 120o] U [240o, 360o] for $$0^o\leq \theta \leq 360^o$$

How to continue

Thanks

lanedance
Homework Helper

so you're pretty much there,

first though, the way to visualise this is to consider all the vectors pointing in the same direction, theta = 0. this is where |v| = 3

as the angle is increased, imagine the vectors spreading something like a flower opening, keeping the same angle between each, with |v| decreasing. The maximum angle occurs when they are all in a plane, theta = 120, and |v| = 0. Agreeing with the first range of your solution.

I also think you only need to consider upto 120 (solutions for 120<theta<= 180 do not exist, and above 180 you can just measure the angle the other way)

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Hi lanedance

Ahh I get it now

Thanks a lot for you both !!