What is this vector problem asking?

[reddawg]

Homework Statement

If r = <x, y, z> and r₀ = <x₀, y₀, z₀>, describe the set of all points (x, y, z) such that the magnitude of r – r₀ = 4.

Homework Equations

The Attempt at a Solution

I don't know what this problem is asking to even attempt a possible solution. Please help?

 

[PeroK]

If you start by trying r0 = <0, 0, 0> does that help? Can you see what that means for r?

 

[reddawg]

Trying that I get x² + y² + z² = 16 after finding the magnitude, but I still can't see what that does to r.

 

[D H]

What kind of object does that equation describe?

 

[PeroK]

What if you lose the z and reduce it to two dimensions: $$x^2 + y^2 = 16$$?

 

[reddawg]

Including the z^2, the equation represents a sphere with radius 4. Without the z^2 it's a circle.

 

[PeroK]

Good. And, if r_0 is the point <x0, y0, z0>?

 

[reddawg]

I'm guessing that the reverse would be true making the components of r_0 negative, however squaring those would make that irrelevant. This is where I'm not following.

 

[PeroK]

Think geometrically.

 

[reddawg]

So r_0 is the center/origin of the sphere/circle?

 

[PeroK]

Yes, that's it. $$|r - r_0| = 4$$ is the vector equation for a sphere of radius 4, centred at r0.

It's equivalent to $$(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = 16$$

 

[reddawg]

Wow I way over thought this.

Thanks PeroK.