Find Equation of Line Passing Through $(1, -2, -3)$ & Perpendicular to Plane

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In summary, the equation of the line that passes through (1, -2, -3) and is perpendicular to the plane 3x-y-2z+4=0 is given by the vector equation l(t)=(1, -2, -3)+t(3, -1, -2). This is the most efficient way to find the equation and was well-formulated. The conversation ends on a positive note with the person thanking the expert for their help.
  • #1
mathmari
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Hello! :eek:

Find the equation of the line that passes through $(1, -2, -3)$ and that is perpendicular to the plane $3x-y-2z+4=0$.

Can we formulate it as followed??

The perpendicular vector of the plane is $\overrightarrow{v}=(3, -1, -2)$.

Since the line that we are looking for is also perpendicular to the plane, we have that the line is parallel to the vector $\overrightarrow{v}$.

Therefore, the line is $$\overrightarrow{l}(t)=(1, -2, -3)+t \overrightarrow{v}$$

Is it correct?? (Wondering)

Could I improve something at the formulation ?? (Wondering)
 
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  • #2
Hi,

It's OK, and I think is the "cheapest" way to find the equation, good work! :cool:
 
  • #3
Fallen Angel said:
Hi,

It's OK, and I think is the "cheapest" way to find the equation, good work! :cool:

Great! Thank you very much! (Smile)
 

1. What is the equation of a line passing through a given point and perpendicular to a given plane?

The equation of a line passing through a given point and perpendicular to a given plane is called the normal form of the line. It can be represented as x = x0 + at, y = y0 + bt, z = z0 + ct, where (x0, y0, z0) is the given point, and (a, b, c) is the normal vector of the plane.

2. How do you find the normal vector of a plane?

The normal vector of a plane can be found by taking the cross product of two non-parallel vectors in the plane. These vectors can be obtained by subtracting any two points on the plane. The resulting vector will be perpendicular to the plane and can be used as the normal vector in the equation of the line.

3. Can the line passing through a point and perpendicular to a plane intersect the plane?

No, the line passing through a point and perpendicular to a plane will not intersect the plane. This line and the plane will be parallel to each other, and therefore, will not have any points in common.

4. Is the equation of a line passing through a point and perpendicular to a plane unique?

Yes, the equation of a line passing through a point and perpendicular to a plane is unique. This is because the normal vector of the plane is unique, and it determines the direction of the line passing through the given point.

5. How can I check if a line is perpendicular to a plane?

To check if a line is perpendicular to a plane, you can take the dot product of the line's direction vector and the plane's normal vector. If the dot product is equal to 0, then the line is perpendicular to the plane. If the dot product is not equal to 0, then the line and the plane are not perpendicular.

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