Determine if the planes in each pair are parallel and distinct or coincident.
a) 5x - 2y + 4z = 7
5x - 2y + 4z = -3
b) 7x - 3y - z = 9
21x - 9y - 3z = 27
The coefficients of the left-hand side determine the vector orthogonal to the plane.
Reduce each plane so that the left-hand side are identical (already are for (a)).
If the left-hand sides cannot be made identical, the planes are not parallel.
If they are parallel, look at the right hand side.
If the RHS are distinct, so are the planes. If the right-hand sides are identical, the two planes are coincident.
I don't understand it still.. So is a) parallel and b) coincidental? Please help! :(
That is correct.
Yes, you've got it right.
In (a), you see clearly that the equations are different, so they represent different planes. In addition, the coefficients of x, y and z are identical, therefore they are parallel, since the orthogonal vector of both planes are (5,-2,4).
For (b), after reduction, both equations become:
7x - 3y - z = 9
so that means there is only one equation that represents both planes, meaning that the two planes are coincident.
If you still have doubts, please post and explain where the doubts are, or what is not clear to you.
Thank you! everything is clear :)
Could you please help me with this question though, I don't understand this one:
Explain why a line in three-space cannot be represented by a scalar equation.
thank you!
Sorry, I do not understand what you mean by a scalar equation.
A line in three-space is defined by the intersection of two planes, each of which is of the form:
Ax+By+Cz+D=0
A line can also be defined in parametric form, which is basically a point to define a position in space, followed by a vector to define the direction, multiplied by a scale factor for the distance from , such as
(x,y,z)=(x0,y0,z0)+t(a,b,c)
where (a,b,c) is the direction vector, and t is a parameter.
thank you! :)
You're welcome!
To determine if two planes are parallel and distinct or coincident, we need to compare their equations.
a) For the planes given by the equations 5x - 2y + 4z = 7 and 5x - 2y + 4z = -3, we can see that the coefficients of x, y, and z are the same for both equations. This suggests that the planes may be parallel. To confirm, we need to check if the constant terms are also equal.
In this case, the constant term in the first equation is 7, while the constant term in the second equation is -3. Since the constant terms are different, the planes are not coincident. Therefore, the planes are parallel and distinct.
b) For the planes given by the equations 7x - 3y - z = 9 and 21x - 9y - 3z = 27, again, we observe that the coefficients of x, y, and z are the same for both equations. This indicates that the planes may be parallel. To verify, we should compare the constant terms.
The constant term in the first equation is 9, while the constant term in the second equation is 27. Since the constant terms are different, the planes are not coincident. Hence, the planes are parallel and distinct.
In summary:
a) The planes 5x - 2y + 4z = 7 and 5x - 2y + 4z = -3 are parallel and distinct.
b) The planes 7x - 3y - z = 9 and 21x - 9y - 3z = 27 are also parallel and distinct.