Find Equation of Plane Perpendicular to Line $l(t)$

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In summary, the conversation discusses finding the equation of a plane that is perpendicular to a given line and passes through a given point. Two methods are mentioned and the second method is confirmed to be correct. The equation of a plane is also discussed and the process of finding the equation in this specific case is summarized.
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evinda
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Hello! (Wave)

If we want to find the equation of the plane that is perpendicular to the line $l(t)=(-1,-2,3)t+(0,7,1)$ and passes through $(2,4,-1)$.

If (x,y,z) is a point of the line then it is of the form (-t,-2t+7,3t+1).

We get that $t=-x=\frac{7-y}{2}=\frac{z-1}{3} \Rightarrow -2x=7-y=\frac{2z-2}{3} \Rightarrow -6x=21-3y=2z-2 $

Can we find in that way the equation of the plane?

Also, is the following way also right to find the desired equation?

For $t=1$ , $a=(-1,5,4)$ where $a \in l(t)$.

Let $(x,y,z)$ be a point of the plane.

Then $(x-2,y-4,z+1)$ belongs to the plane.

Since $l(t)$ is perpendicular to the plane, $a$ is also perpendicular to the plane and since $(x,y,z)$ is a point of the plane, $a$ is perpendicular to $(x,y,z)$. So $a \cdot (x,y,z)=0 \Rightarrow -x+5y+4z=14$
 
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evinda said:
For $t=1$ , $a=(-1,5,4)$ where $a \in l(t)$.

Let $(x,y,z)$ be a point of the plane.

Then $(x-2,y-4,z+1)$ belongs to the plane.
Why?

evinda said:
Since $l(t)$ is perpendicular to the plane, $a$ is also perpendicular to the plane
If you write $a \in l(t)$, it means that $a$ is a point, and a point cannot be perpendicular to anything.

evinda said:
since $(x,y,z)$ is a point of the plane, $a$ is perpendicular to $(x,y,z)$.
Again, points can't be perpendicular.

It is known that the equation of a plane is $Ax+By+Cz+D=0$ where $(A,B,C)$ are the coordinates of a vector perpendicular to the plane. In your case, this vector is $(−1,−2,3)$. It is left to substitute the coordinates of a point on the plane into the equation to find $D$.
 

FAQ: Find Equation of Plane Perpendicular to Line $l(t)$

What is a plane perpendicular to a line?

A plane perpendicular to a line is a plane that intersects the given line at a 90 degree angle, also known as a right angle. This means that the plane and the line are perpendicular to each other, creating a 3-dimensional shape.

How do you find the equation of a plane perpendicular to a line?

To find the equation of a plane perpendicular to a line, you will need to use the cross product of two vectors. The first vector will be the direction vector of the given line, and the second vector will be the normal vector of the plane. The resulting equation will be in the form of Ax + By + Cz = D, where A, B, and C are the coefficients of the normal vector, and D is a constant.

What is the purpose of finding the equation of a plane perpendicular to a line?

The equation of a plane perpendicular to a line is useful in many applications, such as in physics, engineering, and computer graphics. It can be used to determine the orientation of an object or to calculate the intersection of two planes.

Can there be more than one plane perpendicular to a given line?

Yes, there can be an infinite number of planes that are perpendicular to a given line. This is because there are infinite directions in which a plane can be perpendicular to a line, as long as it intersects the line at a right angle.

How does the position of the line affect the equation of the perpendicular plane?

The position of the line does not affect the equation of the perpendicular plane. As long as the line and the plane are perpendicular to each other, the resulting equation will be the same. However, the position of the line may affect the values of the coefficients in the equation.

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