Vectors - Lines and Planes Question

**shiiganB** · April 15th 2010, 10:28 PM

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)

**undefined** · April 16th 2010, 06:07 PM

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

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

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

In this case, write

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

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

This can also be written

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

In other words, we have a parametric equation described by

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

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

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

$t=5$

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

$D^2 = 5^2 + 5^2 + 5^2$

$D = 5\sqrt{3}$

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

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

Thread: Vectors - Lines and Planes Question

LinkBack

Thread Tools

Display

Vectors - Lines and Planes Question

Similar Math Help Forum Discussions

Distance between 2 parametric lines - Vectors/Lines and Planes

Help with Lines/Planes/Vectors

Vectors, lines, and planes

Equation of lines / planes (vectors)

vectors.lines.planes

Search Tags