problem involving vectors, planes and points

question:

1. according to the 'flat earth club', Earth is a plane described by the equation: x + y + z = 18.

also according to the flat earth club, earth will be destroyed on 31.8.09 by an explosion that will spontaneously occur at the so-called ' armageddon point' A = (1,1,1) in space. it so happens that melbourne (considered as a point) will be the first place on earth that is destroyed by this explosion.

a) calculate the coordinates of melbourne

b) what is the distance between A(the armageddon point) and melbourne. what is the distance between A and the earth?

2. a) according to the flat mars society. mars is also a plane, given by the equation: 4x +3y - z = -3.

find a parametric equation of the line in which earth (as in question 1) intersects mars.

b) how far away is canberra, given by the point (-5,10,13), from this line?

3. Two perfectly round pieces of rock are hurtling through space. The first, Superman’s

holiday asteroid, has radius 0.3, and is traveling on the line given by the parametric

equation

x(t) = 2 + t, y(t) = −1 − t, z(t) = t.

The second rock, made of kryptonite and set in motion by one of Superman’s enemies,

has radius 0.1 and is traveling on a line given by the parametric equation

x(s) = 3 − s, y(s) = 1, z(s) = 1 + s.

Calculate the distance between the two lines (8 marks) and use this distance to prove

that Superman has nothing to worry about. (2 marks)