Distance between planes | Vectors and spaces | Linear Algebra | Khan Academy

TL;DR

The video explains how to find the distance between two parallel planes using the equation of the planes and a point on one of the planes. The absolute value of the constant term in the equation represents the distance between the planes.

Transcript

If the distance between the plane Ax-2y+z = d and the plane containing the lines, and they give us two lines here in three-dimensions, if that distance is square-root of 6, then the absolute value of d is... So let's think about it for a little bit. They're talking about the distance between this plane and some plane that contains these two line. S... Read More

Key Insights

• ✈️ The distance between two parallel planes in three dimensions can be calculated using the equation of one of the planes and a point on that plane.
• ✈️ Two non-collinear points on a plane can be used to find two vectors on the plane, which can then be used to calculate a normal vector.
• ✈️ The absolute value of the constant term in the plane equation represents the distance between the plane and the origin of the coordinate system.
• 🍉 The absolute value of the constant term can also be used to determine the distance between two parallel planes.
• ✈️ The distance between the two planes will be zero if the planes intersect.
• ✈️ The planes must be parallel in order to accurately calculate the distance between them.
• 🤪 The ratio of the coefficients of the x, y, and z terms in the plane equations determines if the planes are parallel.
• ✈️ The equation of a plane can be found using a point on the plane and the normal vector. The dot product of the normal vector and an arbitrary vector on the plane is equal to zero.

Questions & Answers

Q: How can the distance between two planes be calculated?

The distance between two parallel planes can be calculated by finding the equation of one of the planes and using a point on that plane to determine the distance to the other plane.

Q: What condition must be met for the distance between two planes to be non-zero?

The two planes must be parallel to each other for the distance between them to be non-zero. If the planes intersect, the distance will be zero.

Q: What is the significance of the absolute value of the constant term in the plane equation?

The absolute value of the constant term (usually represented as 'd') in the plane equation represents the distance between the plane and the origin of the coordinate system.

Q: How can the equation of a plane be derived using two points on the plane?

By using two non-collinear points on the plane, you can find two vectors on the plane. Taking the cross product of these vectors will yield a normal vector, which can be used to derive the equation of the plane using the point-normal form.

Summary & Key Takeaways

• The video discusses the concept of finding the distance between two parallel planes in three dimensions.

• It explains that for the planes to have a non-zero distance, they must be parallel.

• The video demonstrates the process of finding the equation of one of the planes and using a point on that plane to calculate the distance to the other plane.