Find parametric equations and symmetric equations for the line. The line

achieverh3

achieverh3

Answered question

2021-12-03

Find parametric equations and symmetric equations for the line.
The line of intersection of the planes
x+2y+3z=1
and
xy+z=1

Answer & Explanation

Uersfeldte

Uersfeldte

Beginner2021-12-04Added 20 answers

Step 1 
Formula: 
Parametric equations are given by 
x=x0+a task

 

y=y0+b task z=z0+c task
Symmetric equations are given by 
xx0a=yy0b=zz0c 
Explanation: 
To find the parametric and symmetric equations the given equations then, 
Consider a point <x, y, z> must satisfies the both equations 
x+2y+3y=1 and xy+z=1 
1) x+2y+3y=1 
2) xy+z=1 
Step 2 
For finding the values of x0, y0, z0 
To find the intersection point: 
Let x=0 or y=0 or z=0 
Consider here z=0 then 
Equation (1) and (2) implies, 
3) x+2y=1 
4) xy=1 
Now solve these two simultaneous equations 
Multiply equation (4) by 2 and add it in equation (3) 
Therefore (4) implies, 2x2y=2 
x+2y=1 
+2x2y=2 
3x=3 
x=1 
Now put x=1 in Equation (3) 
1+2y=1 
2y=11 
2y=0 
y=0 
Therefore the point of intersection is <1, 0, 0<x0, y0, z0> respectively. 
Step 3 
For finding the values of a, b, c : 
Find the vector that is perpendicular to the planes 
1) x+2y+3y=1 
2) xy+z=1 
P^1=1, 2, 3 and P^2=1, 1, 1 
P^1×P^2=[i^j^k^123111] 
=i^[2311]j^[1311]+k^[1211] 
=i^[2(1)3(1)]j^[1(1)3(1)]+k^[1(1)2(1)] 
 

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