Journey Through a Sphere

Geometry Level 5

In the $xyz$ coordinate system, suppose we begin at $(x,y,z) = (2,3,4)$ and head off in a direction described by the vector $(vx,vy,vz) = (-3,-6,-7)$ .

Along the way, there are two points at which we intersect a unit sphere centered at the origin. What is the distance between the two points of intersection?

Details and Assumptions:
Give your answer to two decimal places.

The answer is 1.75.

1 solution

Michael Mendrin
Sep 20, 2016

Define the line as a parametric function of $t$

$x=2-3t$
$y=3-6t$
$z=4-7t$

Expanding ${x}^{2}+{y}^{2}+{z}^{2}-1=0$ gets us the quadratic equation to solve for $t$

$94{t}^{2}-104t+28=0$

the solutions being

$t=\dfrac{1}{47}\left(26-3\sqrt{2}\right)$
$t=\dfrac{1}{47}\left(26+3\sqrt{2}\right)$

From this, we can find the $2$ points on the sphere that the vector passes through, the distance between them working out to a neat $\dfrac{12}{\sqrt{47}}=1.7503798...$

Journey Through a Sphere

The answer is 1.75.

1 solution

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