# Vectors - Lines and Planes Question

• Apr 15th 2010, 10:28 PM
shiiganB
Vectors - Lines and Planes Question
According to an astronomy club, Earth is a plane described by the equation
x + y + z = 18.

The club also stated that Earth will be destroyed by an explosion that will spontaneously occur at point A = (1, 1, 1). Note that point A is not a point on Earth, it is a point in space that will explode and will impact Earth.

a/ Calculate the co-ordinate of the first point that is going to be destroyed by this explosion.

b/ What is the distance between point A and the first point that is going to be destroyed (mentioned in a)? (Hint: Shortest Distance)
• Apr 16th 2010, 06:07 PM
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Quote:

Originally Posted by shiiganB
According to an astronomy club, Earth is a plane described by the equation
x + y + z = 18.

The club also stated that Earth will be destroyed by an explosion that will spontaneously occur at point A = (1, 1, 1). Note that point A is not a point on Earth, it is a point in space that will explode and will impact Earth.

a/ Calculate the co-ordinate of the first point that is going to be destroyed by this explosion.

b/ What is the distance between point A and the first point that is going to be destroyed (mentioned in a)? (Hint: Shortest Distance)

We want to find the line perpendicular to the plane passing through the point; this will tell us the minimal distance between the point and the plane.

For a plane $\displaystyle ax + by + cx + d = 0$, we have normal vector $\displaystyle \vec n = (a, b, c)$.

The vector equation of the line passing through point $\displaystyle P_0$ with direction vector $\displaystyle \vec v$ is

$\displaystyle \vec r(t)=P_0+t\cdot \vec v$

In this case, write

$\displaystyle \vec n = (1, 1, 1)$

$\displaystyle \vec r(t)=(1,1,1)+t\cdot(1,1,1)$

This can also be written

$\displaystyle \vec r(t)=(1+t,1+t,1+t)$

In other words, we have a parametric equation described by

$\displaystyle (x,y,z)=(1+t,1+t,1+t)$

Combine with $\displaystyle x+y+z=18$ to get

$\displaystyle (1+t)+(1+t)+(1+t)=18$

$\displaystyle t=5$

So the point that gets hit first is $\displaystyle (6,6,6)$, and to find the distance, use the Pythagorean theorem (generalized for three dimensions):

$\displaystyle D^2 = 5^2 + 5^2 + 5^2$

$\displaystyle D = 5\sqrt{3}$

Note that the line we found earlier also passes through the origin, so it could be expressed more concisely as

$\displaystyle \vec r(t)=(t,t,t)$