Determining the distances between points and planes

In summary, the user completed a question and requested a check on their solution. They also questioned whether they should reduce their answer. Another user responded by confirming that the formula used was correct and provided a detailed explanation of how to calculate the distance between a point and a line. They also suggested drawing a sketch to better understand the concept.
  • #1
ttpp1124
110
4
Homework Statement
Hi! I finished the question, can someone check? Also, should I be reducing my answer?
Relevant Equations
n/a
-
 
Last edited:
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  • #2
ttpp1124 said:
Homework Statement:: Hi! I finished the question, can someone check? Also, should I be reducing my answer?
Relevant Equations:: n/a

https://www.physicsforums.com/attachments/260124
Looks fine to me. The formula you used apparently was
##d = |\vec v|\sin(\theta)##, where ##\vec v## is the vector from a point on the line Q(6, 0, 1) and P(1, -5, 2), and ##\theta## is the angle between ##\overrightarrow{QP}## and the direction vector for the line, <3, 1, 2>.
The magnitude of the cross product of ##\vec u## and ##\vec v##, is ##|\vec u \times \vec v| =|\vec u||\vec v|\sin(\theta)##. With a bit of algebra, you get ##d = \frac{|\vec u \times \vec v|}{|\vec u|}##.
I don't see that your final fraction will simplify very much. The only common factor of 390 and 14 is a single factor of 2.

I also don't see any simple way of checking, other than using the dot product to get the projection of ##\overrightarrow{PQ}## onto the line's direction vector (which will be the base of the right triangle), and then using Pythogoras to get the height of the triangle (which will be the distance from the line to the point.

Something I think would be useful would be to draw a sketch of a line and the given point, and derive either of the formulas for distance. This is more useful, IMO, than merely memorizing a canned formula.
 
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1. How do you determine the distance between a point and a plane?

The distance between a point and a plane can be calculated by finding the perpendicular distance from the point to the plane. This can be done by using the formula d = |ax + by + cz + d| / √(a^2 + b^2 + c^2), where (x, y, z) is the coordinates of the point and ax + by + cz + d = 0 is the equation of the plane.

2. Can the distance between a point and a plane be negative?

No, the distance between a point and a plane is always positive. This is because the perpendicular distance is always measured in a positive direction from the point to the plane.

3. What is the difference between the distance from a point to a plane and the distance between two planes?

The distance from a point to a plane is the shortest distance between the point and any point on the plane. The distance between two planes, on the other hand, is the shortest distance between any two points on the two planes. This distance is measured along a line that is perpendicular to both planes.

4. How do you determine if a point lies on a plane?

To determine if a point (x, y, z) lies on a plane with equation ax + by + cz + d = 0, simply substitute the coordinates of the point into the equation. If the resulting value is 0, then the point lies on the plane. If the resulting value is not 0, then the point does not lie on the plane.

5. Is there a way to calculate the distance between a point and a plane without using the formula?

Yes, there is another method to calculate the distance between a point and a plane. This method involves finding the projection of the point onto the plane, and then calculating the distance between the projected point and the original point. However, this method can be more complicated and may require more advanced mathematical concepts.

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