Representing the following line using the symmetric form and parametric equations {█(x+y+z+1=0@2x-y+4z+4=0)┤ .
时间: 2023-12-28 13:06:15 浏览: 28
To represent the given line using the symmetric form, we can solve the two equations simultaneously to eliminate the variables x, y, and z.
x + y + z + 1 = 0
2x - y + 4z + 4 = 0
From the first equation, we can write:
x = -y - z - 1
Substituting this value of x in the second equation, we get:
2(-y-z-1) - y + 4z + 4 = 0
Simplifying, we get:
-3y + 2z = -6
or, 3y - 2z = 6
This is the symmetric form of the line.
To represent the line using parametric equations, we can choose any two variables to be the parameters, say y and z. Then we can write:
y = t
z = s
Substituting these values in the equation 3y - 2z = 6, we get:
3t - 2s = 6
or, s = (3t - 6)/2
Substituting these values of y and z in the equation x + y + z + 1 = 0, we get:
x = -y - z - 1
or, x = -t - (3t - 6)/2 - 1
Simplifying, we get:
x = (-5t + 4)/2
Therefore, the parametric equations of the line are:
x = (-5t + 4)/2
y = t
z = (3t - 6)/2,
where t and s are any real numbers.