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

Hint: 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.