Determine whether the following pairs of planes are coincident, parallel and distinct or neither.
a) x + 3y - z - 2 = 0 and 2x + 6y - 2z - 8 = 0
TEXTBOOK ANSWER: parallel and distinct
Please provide a step by step solution so I can understand the question...
Thanks.
very simple
notice that if you divide the second equation by 2 you get
x + 3y - z - 4 = 0
compare this with the first to see that they have the same normal, so they are parallel.
Since their constants are differenct they must form two parallel planes.
I call that the "floor-ceiling" situation.
To determine whether the given pairs of planes are coincident, parallel and distinct, or neither, we need to compare their coefficients.
The general equation of a plane in 3D space is given by:
Ax + By + Cz + D = 0
For the first pair of planes:
Plane 1: x + 3y - z - 2 = 0
Plane 2: 2x + 6y - 2z - 8 = 0
We can rewrite the equations in the general form by rearranging the terms:
Plane 1: x + 3y - z = 2 (dividing by -1)
Plane 2: 2x + 6y - 2z = 8 (dividing by -2)
Now, we have the following coefficient matrices:
Plane 1: [1, 3, -1]
Plane 2: [2, 6, -2]
To determine whether the planes are coincident, we need to check if the ratio of the coefficients of x, y, and z is the same for both planes. If they are the same, the planes are coincident.
To determine whether the planes are parallel, we need to check if the ratio of the coefficients of x, y, and z in one plane is proportional to the corresponding ratio in the other plane. If they are proportional, the planes are parallel. If the ratios are not proportional, the planes are distinct.
Now, let's compare the coefficients:
For x:
The ratio of the coefficients is 1/2.
For y:
The ratio of the coefficients is 3/6 or 1/2.
For z:
The ratio of the coefficients is -1/-2 or 1/2.
Since the ratios for all three variables are equal, the planes are parallel. However, they are not coincident because the constant terms are different (2 and 8).
Therefore, the given pair of planes is parallel and distinct.