# Vectors - Parametric Equations

• Apr 15th 2010, 10:35 PM
shiiganB
Vectors - Parametric Equations
Earth is described by the equation : x + y + z = 18
Mars is described by the equation: 4x + 3y - z = -3

Earth and Mars intersect each other

a/ find a parametric equation of the line in which earth intersect mars

b/ The coordinate of Spring Lake is given by the point (-5, 10, 13)
How far away from spring lake from the line mentioned in (a)?
• Apr 16th 2010, 07:23 AM
Soroban
Hello, shiiganB!

Astronomically, the problem makes no sense,
. . but I'll solve it anyway.

Quote:

$\displaystyle \begin{array}{ccccc}\text{Earth is described by the equation:} & x + y + z &=& 18 & [1] \\ \text{Mars is described by the equation:} & 4x + 3y - z &=& \text{-}3& [2] \end{array}$

Earth and Mars intersect each other.

(a) Find a parametric equation of the line in which Earth intersect Mars.

Add [1] and [2]: .$\displaystyle 5x + 4y \:=\:15 \quad\Rightarrow\quad x \:=\:3-\tfrac{4}{5}y$

Substitute into [1]: .$\displaystyle \left(3-\tfrac{4}{5}y\right) + y + z \:=\:18 \quad\Rightarrow\quad z \:=\:15-\tfrac{1}{5}y$

. . We have: .$\displaystyle \begin{Bmatrix}x &=& 3-\frac{4}{5}y \\ y &=& y \\ z &=& 15-\frac{1}{5}y \end{Bmatrix}$

On the right, replace $\displaystyle y$ with the parameter $\displaystyle t.$

. . Therefore: . $\displaystyle \begin{Bmatrix}x &=& 3 - \frac{4}{5}t \\ y &=& t \\ z &=& 15 - \frac{1}{5}t \end{Bmatrix}$

Quote:

(b) The coordinates of Spring Lake is given by the point (-5, 10, 13).
How far away from Spring Lake is the line mentioned in (a)?

A strange (silly!) question . . .

Spring Lake could be a ski resort located on Mars or Jupiter or . . .

Ignoring that possibility, Spring Lake is probably located on Earth,
. . in which case, it is on the line mentioned in (a).

. . The distance is zero.

To check, try $\displaystyle t = 10.$