Question

How do you find the vector parametrization of the line of intersection of two planes $2x - y - z = 5$ and $x - y + 3z = 2$?

Answer 1

We calculate the value of y from the first equation and substitute the value of y in the second equation. Calculate the value of z in terms of x and then use that value to calculate the value of y in terms of x by substituting the value of z in the equation of y. In the end put the value of x as a variable and according to that write all the other values in terms of the same variable.

• Parameterization of an equation means that we write the values of x, y and z in terms of a free variable such that any change in the free variable brings change in the values of x, y and z.

Complete step-by-step answer:
We are given equations for two planes
$2x - y - z = 5$ … (1)
$x - y + 3z = 2$ … (2)
We shift y from left side to right side of the equation (1)
$\Rightarrow 2x - z - 5 = y$ … (3)
Substitute the value of y from equation (3) in equation (2)
$\Rightarrow x - (2x - z - 5) + 3z = 2$
$\Rightarrow x - 2x + z + 5 + 3z - 2 = 0$
$\Rightarrow - x + 4z + 3 = 0$
Shift all values except 4z to right hand side of the equation
$\Rightarrow 4z = x - 3$
Divide both sides of equation by 4
$\Rightarrow z = \dfrac{x}{4} - \dfrac{3}{4}$ … (4)
Substitute the value of z from equation (4) in equation (3)
$\Rightarrow y = 2x - \left( {\dfrac{x}{4} - \dfrac{3}{4}} \right) - 5$
$\Rightarrow y = 2x - \dfrac{x}{4} + \dfrac{3}{4} - 5$
Take LCM of similar terms
$\Rightarrow y = \dfrac{{8x - x}}{4} + \dfrac{{3 - 20}}{4}$
$\Rightarrow y = \dfrac{{ - 7x}}{4} + \dfrac{{ - 17}}{4}$ … (5)
Now we have the values of y and z in terms of x
Let us assume the parameter as ‘t’. Put $x = t$as the perimeter then
$x = t$
$y = \dfrac{{ - 7}}{4}t - \dfrac{{17}}{4}$
$z = \dfrac{1}{4}t - \dfrac{3}{4}$

$\therefore$Vector parameterization of the line of intersection is $x = t$;$y = \dfrac{{ - 7}}{4}t - \dfrac{{17}}{4}$;$z = \dfrac{1}{4}t - \dfrac{3}{4}$.

Note:
Many students make the mistake of using the intercept form to calculate the value of x, y and z as constant values which will only give a solution but we need a line of intersection of the planes. We can choose to find the value of any two variables in terms of one same variable.