How to Calculate Distance from a Point to a Plane

TL;DR

To calculate the distance from a point to a plane, use the formula: distance = |(ax1 + by1 + cz1 + d) / √(a² + b² + c²)|, where (a, b, c) are the normal vector components and (x1, y1, z1) are the coordinates of the point. The absolute value ensures the distance is positive.

Transcript

in this video we're going to talk about how to calculate the distance from a point to a plane now before we actually solve the problem let's talk about how we can derive the formula so let's say we have some plane represented by this picture and a point above that plane let's call that point p1 so our goal is to find the distance between that point... Read More

Key Insights

• 🔯 The formula for calculating the distance between a point and a plane involves finding the dot product of the normal vector and the vector between the point and the point on the plane.
• ❓ The scalar projection of the vector onto the normal vector is used to derive the formula.
• 😥 The distance between a point and a plane should always be positive.
• 🪡 Memorizing the formula is not necessary, as it can be derived and applied when needed.
• 😥 The formula requires the coordinates of the given point, the point on the plane, and the components of the normal vector.
• 👻 Plugging in the values into the formula allows for the calculation of the distance.
• ❎ The derived formula accounts for negative values of d, ensuring the correct result.

Questions & Answers

Q: What is the goal when calculating the distance between a point and a plane?

The goal is to find the shortest distance between the point and the plane.

Q: How is the formula for calculating the distance derived?

The formula is derived by finding the scalar projection of the vector between the point on the plane and the given point onto the normal vector of the plane.

Q: Why do we use the absolute value of the dot product in the formula?

To ensure that the distance between the point and the plane is positive, we take the absolute value of the dot product.

Q: How can the formula be applied to calculate the distance?

By plugging in the coordinates of the given point, the point on the plane, and the components of the normal vector into the formula, the distance can be calculated.

Summary & Key Takeaways

• The video explains how to calculate the distance from a point to a plane by deriving a formula.

• The formula involves finding the dot product of the normal vector of the plane and the vector between the point on the plane and the given point.

• By plugging in the appropriate values into the formula, the distance between the point and the plane can be calculated.