How To Find The Equation of a Plane Given a Point and Perpendicular Normal Vector | Summary and Q&A

TL;DR

Learn how to find the equation of a plane using a given point and a vector that is perpendicular to the plane.

Key Insights

• ✈️ To define a plane, you need a point on the plane and a vector that is perpendicular to it.
• 🫥 The dot product of the normal vector and any vector on the plane is zero.
• ✈️ The equation of a plane can be found using the formula A(X - X0) + B(Y - Y0) + C(Z - Z0) = 0.
• 😥 A, B, and C represent the components of the normal vector, while X0, Y0, and Z0 are the coordinates of the point on the plane.
• 🔚 The equation of a plane is in the form Ax + By + Cz = D, where D is a constant term.
• ✈️ Understanding how to find the equation of a plane is crucial in geometry and physics applications.
• 🫥 The dot product allows us to determine whether two vectors are perpendicular or not.

Transcript

in this video we're going to talk about how to find the equation of a plane using a point and a vector that's perpendicular to the plane in fact those are the two things that we need in order to define a plane let's start with a picture so here we have a 3d coordinate system this is going to be x y and z and let's draw a plane somewhere in this reg... Read More

Questions & Answers

Q: What are the two things needed to define a plane?

A point on the plane and a vector that is perpendicular to it.

Q: How can you determine if two vectors are orthogonal or perpendicular?

If the dot product of the two vectors equals zero, they are orthogonal or perpendicular to each other.

Q: How do you find the equation of a plane using a point and normal vector?

Using the formula A(X - X0) + B(Y - Y0) + C(Z - Z0) = 0, where A, B, and C are the components of the normal vector, and X0, Y0, and Z0 are the coordinates of the point.

Q: Can you provide an example of finding the equation of a plane?

Let's say A is 3, B is 6, C is 5, X0 is 2, Y0 is -5, and Z0 is 3. Plugging these values into the formula, we get 3x + 6y + 5z = -9 as the equation of the plane.

Summary & Key Takeaways

• To define a plane, you need a point on the plane (P0) and a vector (N) that is perpendicular to it.

• The dot product of the normal vector (N) and the vector from P0 to any other point on the plane (R - R0) equals zero.

• Using the formula, A(X - X0) + B(Y - Y0) + C(Z - Z0) = 0, you can find the equation of the plane.