Equation of a plane that passes thorugh the line of intersection of [answer check]

In summary, the equation of the plane passing through the line of intersection of 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0 and also through point A(1, -3, 2) is (72/13)x + (1/13)y - (3/13)z - (63/13) = 0.
  • #1
lamerali
62
0
Find the equation of the plane that passes through the line of intersection of the planes 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0 and passes through A(1, -3, 2).

I have answered the question but I am not sure it is correct; if someone can review it their comments would be appreciated!
Thank you!


The plane that passes through the line of intersection of the two planes has the equation

4x – 3y – z – 1 + k(2x +4y + z – 5) = 0

Substitute the point A(1, -3, 2) into the equation to find k

4(1) – 3(-3) – 2 – 1 + k (2(1) + 4(-3) + 2 – 5) = 0
10 + k(-13) = 0
k = 10 / 13

substitute the value of k into the equation of the plane

4x – 3y – z – 1 + (10/13)(2x +4y + z – 5) = 0
4x – 3y – z – 1 + (20/13)x + (40/13)y + (10/13)z – (50/13) = 0

Therefore the equation of the plane that passes through the line of intersection of the two planes is

(72/13)x + (1/13)y – (3/13)z – (63/13) = 0
 
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  • #2


Yup it looks ok.
 
  • #3


Great! :D Thank you!
 

FAQ: Equation of a plane that passes thorugh the line of intersection of [answer check]

Question 1: What is the equation of a plane that passes through the line of intersection of two planes?

The equation of a plane that passes through the line of intersection of two planes can be found by first finding the cross product of the normal vectors of the two planes. This cross product will give you the direction vector of the line of intersection. Next, choose any point on the line of intersection and use it as the point in the point-normal form of a plane equation. The resulting equation will be the equation of the plane that passes through the line of intersection.

Question 2: How do I find the normal vectors of two intersecting planes?

To find the normal vectors of two intersecting planes, simply take the coefficients of the x, y, and z terms of each plane's equation and put them into a vector. The resulting vectors will be the normal vectors of the planes. Keep in mind that these normal vectors will be perpendicular to their respective planes.

Question 3: Can the equation of a plane passing through the line of intersection of two planes be simplified?

Yes, the equation of a plane passing through the line of intersection of two planes can be simplified. One way to simplify it is to use the direction vector of the line of intersection as the normal vector of the plane, resulting in a simpler equation. Another way is to rearrange the equation to put it in standard form, which can make it easier to visualize and work with.

Question 4: How do I know if the resulting plane from the intersection of two planes is unique?

The resulting plane from the intersection of two planes will be unique if the two planes are not parallel to each other. This means that they have different normal vectors and thus will have a unique line of intersection. However, if the two planes are parallel, they will not intersect and there will not be a unique plane passing through their line of intersection.

Question 5: Is there any special case when finding the equation of a plane passing through the intersection of two perpendicular planes?

Yes, when finding the equation of a plane passing through the intersection of two perpendicular planes, the resulting plane will have a normal vector that is the cross product of the normal vectors of the two planes. This is because the line of intersection of two perpendicular planes will be parallel to both planes' normal vectors, making their cross product the direction vector of the line of intersection.

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