Asked by J
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
a) 5x - 2y + 4z = 7
5x - 2y + 4z = -3
b) 7x - 3y - z = 9
21x - 9y - 3z = 27
Answers
Answered by
MathMate
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.
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.
Answered by
J
I don't understand it still.. So is a) parallel and b) coincidental? Please help! :(
Answered by
Steve
That is correct.
Answered by
MathMate
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.
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.
Answered by
J
Thank you! everything is clear :)
Answered by
J
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!
Explain why a line in three-space cannot be represented by a scalar equation.
thank you!
Answered by
MathMate
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.
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.
Answered by
J
thank you! :)
Answered by
MathMate
You're welcome!
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