# How Do You Find a Plane Orthogonal to a Given Line?

• Whatupdoc
In summary: This vector is denoted by \vec{P}_1 - \vec{P}_2. The two components of \vec{P}_1 - \vec{P}_2 are (2x, -5y, -2z). In summary, the equation of a plane through the point (-1, -2, -3) which is orthogonal to the line x=5+2t,y=-3-5t,z=2-2t is N*r = n*r_0. The coefficient of x is 2.
Whatupdoc
Find an equation of a plane through the point (-1, -2, -3) which is orthogonal to the line x=5+2t,y=-3-5t,z=2-2t
in which the coefficient of x is 2.

______________________________=0

i don't get this problem at all, but here's what i came up with after sitting here at the computer for a long time attempting to do this problem.

okay i know that by theorem, The vector a X b is orthogonal to both a and b.

a = <-1,-2,-3>
b= <2,-5,2>

that's all i came up, i attempted many different ways, but it doesn't make sense at all, can someone lend me a hand?

Ok, here's a plan: Find out the general form of a vector parallel to the line. (it will be a function of two parameters; say you labeled them t_1 and t_2.)Write out the form of a vector that lies in the plane (it will be a function of x,y & z). Then take the dot product of those two vectors and set it equal to 0. This is the equation of your plane. Now chose a particular vector parrarel to the line such that the coeff. of x is 2... that is to say, set t_1 and t_2 such that the coeff. of x is 2.

I understand that this might sound very confusing but try your best to progress. If you get stuck, or if after thinking very hard about it, you still can't find a way, I'll help you some more. Good luck!

N*r = n*r_0 is the equation I am looking for right?

i am totally lost here, book doesn't help much either, can you help me start it.

What are N,r,n and r_0 ?!

Have you done more simple similar problems before? If you're new to this kind of problem then this one is a rought introduction IMO!

Let's start with the first step of my plan. Let's find the general form of a vector parallel to the line. The equation of the line is written in parametric form. It means that for any value of t, the point (x,y,z)=(5+2t,-3-5t,-2-2t) is on the line. Ok, so let's find two points on the line by setting t = t_1 and t = t_2. Our two points are $\vec{P}_1 = (5+2t_1,-3-5t_1,-2-2t_1)$ and $\vec{P}_2 = (5+2t_2,-3-5t_2,-2-2t_2)$. By means of a simple drawing, you can convince yourself that $\vec{P}_1 - \vec{P}_2 = (2(t_1-t_2), -5(t_1-t_2), -2(t_1-t_2))$ is a vector parallel to the line (it is in fact IN/MERGED WITH the line).

Now, make sure you understand every step in this paragraph and do step 2 on your own. Supposing (x,y,z) is a point in the plan, use the same method as above (substraction) to find a vector that is parallel to the plan.

## 1. What is the definition of an orthogonal equation plane?

An orthogonal equation plane is a mathematical representation of a plane that is perpendicular to a given set of axes. This means that the plane intersects each axis at a 90 degree angle.

## 2. How do you find the orthogonal equation of a plane?

To find the orthogonal equation of a plane, you can use the normal vector of the plane and a point on the plane to create a vector equation. Then, you can use this equation to find the orthogonal projection of any point onto the plane.

## 3. What is the significance of an orthogonal equation plane?

An orthogonal equation plane is important in mathematics as it allows for the representation of 3-dimensional objects and their relationships in a 2-dimensional plane. It is also used in physics and engineering to solve problems related to forces and motion in three dimensions.

## 4. Can an orthogonal equation plane be used to solve real-world problems?

Yes, an orthogonal equation plane can be used to solve real-world problems in various fields such as physics, engineering, and architecture. For example, it can be used to determine the shortest distance between a point and a plane, or to find the angle between two intersecting planes.

## 5. Are there any applications of orthogonal equation plane in computer graphics?

Yes, orthogonal equation planes are extensively used in computer graphics to create 3-dimensional objects and scenes. They are used to determine the placement and orientation of objects in a 3D space, and to manipulate and transform them using mathematical operations.

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