Visualizing the plane, 1x+1y+1z = 0

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In summary, the conversation discussed the difficulty in visualizing the traces of the equation x+y+z=0 and how it differs from other equations. It was suggested to graph the equation on paper and calculate points other than the origin to get a better understanding of the plane. The value of d=0 was also mentioned as implying that the plane goes through the origin. The conversation ended with the participants thanking each other for their input.
  • #1
Ocata
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Hello,

I made traces for the equation, x+y+z=0, but they don't seem to connect in an intuitive way as other equations do. For instance, even with x+y+z=1, I can make traces where the 3 lines connect to make a triangle in the first/positive quadrant. But x+y+z=0 has traces that all run through the origin. Not sure how to draw/connect traces for a plane whose traces seem to only intersect at the origin.
 
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  • #3
Ocata said:
Hello,

I made traces for the equation, x+y+z=0, but they don't seem to connect in an intuitive way as other equations do. For instance, even with x+y+z=1,
x + y + z = 1 is a different plane than x + y + z = 0. The two planes are parallel, though, but don't share any points.
Ocata said:
I can make traces where the 3 lines connect to make a triangle in the first/positive quadrant. But x+y+z=0 has traces that all run through the origin. Not sure how to draw/connect traces for a plane whose traces seem to only intersect at the origin.
The origin is a point on your plane. It might help to draw traces in the three coordinate planes. For example, in the x-y plane (where z = 0), the trace is the line x + y = 0, or equivalently, the line y = -x.

To get a three-dimensional view of this plane, calculate two points other than the origin (which is on the plane). Those three points should give you some idea of how the plane looks.
 
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  • #4
Thank you Spinner,

I did graph it on paper but it still didn't make sense because I was tracing through the intercepts and I wasn't able to get a clear idea of the plane. In the equation, d = 0. Now I see that d = 0 implied the plane goes through the origin. Thank you.
 
  • #5
Thank you Mark44.

I have a clear visualization of x + y + z = 0. Your first statement sealed the deal visually for me. Your suggestion to calculate points other than the origin allowed me to realize I can find intercepts other than 0 + y = 0, x + 0 = 0, and z + 0 = 0.

Thank you.
 

1. What does the equation 1x+1y+1z=0 represent in terms of planes?

The equation 1x+1y+1z=0 represents a plane in three-dimensional space. It is known as a general form equation of a plane, where x, y, and z represent the coordinates of any point on the plane. The coefficients 1, 1, and 1 in front of x, y, and z indicate the direction and orientation of the plane. The constant term 0 represents the distance of the plane from the origin.

2. How can one visualize the plane 1x+1y+1z=0?

One way to visualize the plane 1x+1y+1z=0 is to plot points on the plane and connect them to form a flat surface. Another way is to imagine a flat surface passing through the origin and extending infinitely in all directions. This surface would represent the plane 1x+1y+1z=0.

3. What is the normal vector of the plane 1x+1y+1z=0?

The normal vector of the plane 1x+1y+1z=0 is n = (1, 1, 1). This vector is perpendicular to the plane and points in the direction of the plane's normal. The dot product of this normal vector and any vector lying on the plane will always be equal to 0, as seen in the equation 1x+1y+1z=0.

4. How does changing the coefficients in the equation 1x+1y+1z=0 affect the plane?

Changing the coefficients in the equation 1x+1y+1z=0 will affect the direction and orientation of the plane. For example, if the coefficient of x is increased, the plane will tilt towards the x-axis. If the coefficient of z is made negative, the plane will tilt downwards in the z-direction. However, the plane will always pass through the origin and remain flat.

5. Can the equation 1x+1y+1z=0 represent any other shape besides a plane?

No, the equation 1x+1y+1z=0 can only represent a plane. This is because the equation only has three variables (x, y, and z) and no higher-order terms. A plane in three-dimensional space can only be defined by three variables, and the general form of a plane equation will always be of the form ax+by+cz=d, where a, b, and c are the coefficients and d is the constant term. Any equation with more or less than three variables cannot represent a plane in three-dimensional space.

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