# Equation of Normal: M(2,3,1) to Line l

• chmate

#### chmate

Write equation of normal from point $$M(2,3,1)$$ into line:$$l: \frac{x+1}{2}=\frac{y-0}{-1}=\frac{z-2}{3}$$.

Thank you.

What is the relationship between the line and the normal?

Calculate the plane normal to the given line and that passes through 2,3,1

Then calculate the point where the given line and the plane intersect.

This point will enable you, together with the (2,3,1), to calculate the direction of the normal to the given line

marlon

Calculate the plane normal to the given line and that passes through 2,3,1

Then calculate the point where the given line and the plane intersect.

This point will enable you, together with the (2,3,1), to calculate the direction of the normal to the given line

marlon

Let me understand this. I have to create a plane which passes through that point. Another point is point of intersection. But to find that point i must have a direction vector of that plane and I don't have so can you help me with point of intersection. How can i find it?

So, when I have that point I have to create a vector which passes through point (2,3,1) and point of intesection and cross product of vector direction of line and vector I created will create a normal vector of plane. Am I right?

Thank you.

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Let me understand this. I have to create a plane which passes through that point. Another point is point of intersection. But to find that point i must have a direction vector of that plane and I don't have so can you help me with point of intersection. How can i find it?

So, when I have that point I have to create a vector which passes through point (2,3,1) and point of intesection and cross product of vector direction of line and vector I created will create a normal vector of plane. Am I right?

Thank you.

Let's take this one step at the time :

First you need to find a plane that contains the point (2,3,1) and that is normal to the given line (your first equation).

Do you know how to do this ?

marlon

No, can you help me with this and line of intersection?

Thank you.

No, can you help me with this and line of intersection?

Thank you.

let's say that the plane has equatiuon ux+vy+wz+t=0 where (u,v,w) is the normal vector of the plane and t is a constant. We need to find u,v,w and t.

A) find u,v,w
B) find t

For A) i say that (u,v,w) denotes the same direction vector as the line.
Do you know why and if so, what's the direction vector of the line ?

If you have u,v,w then you can use the fact that (2,3,1) lies on the plane to find t

marlon

So the normal vector has the same direction as line. So how to find that vector and t?

So the normal vector has the same direction as line. So how to find that vector and t?

Well, you know the normal vector (u,v,w) and you know one point on the plane

Substitute these two variables in the ux+vy+wz+t=0 and solve for t

marlon