What is the volume and area determined by three points?

In summary, the area of the parallelogram determined by vectors \overrightarrow{PQ} and \overrightarrow{PR} is 9 \sqrt{2} and the volume of the rectangular prism determined by \overrightarrow{PQ}, \overrightarrow{PR}, and the origin O is 81 \sqrt {2}. These calculations were made using the cross product and the dot product.
  • #1
soopo
225
0

Homework Statement


Let P = (2, 2, 0), Q = (0, 4, 1) and R = (-1, 2, 3) in the space [tex] \Re^{3}.
[/tex]
a) Determine the area of the rectangle determined by vectors [tex]\overrightarrow{PQ}[/tex] and [tex]\overrightarrow{PR}[/tex].
b) Determine the volume of the tetrahedral determined by vectors [tex]\overrightarrow{PQ}[/tex] and [tex]\overrightarrow{PR}[/tex], and the origin O, OPQR.

The Attempt at a Solution


a)
[tex]
|\overrightarrow{PQ}| = \sqrt {4 + 4 +1} = 3
[/tex]

[tex]
|\overrightarrow{PR}| = \sqrt {9 + 0 + 9} = 3 \sqrt {2}
[/tex]

[tex]
Area = |\overrightarrow{PQ}| * |\overrightarrow{PR}|
= 9 \sqrt {2}
[/tex]

b)
[tex]
Volume = Area * |\overrightarrow{RO}|
|\overrightarrow{RO}| = \sqrt {1 + 4 + 9}
= \sqrt {14}
[/tex]

The volume is
[tex]
Volume = 9 \sqrt{2} * \sqrt {14}
= 18 \sqrt {7}
[/tex]

Please, comment any mistakes.
 
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  • #2
PR and PQ are not orthogonal, they don't determine a rectangle. Ditto for the second problem. You should be using the cross product and the dot product to solve these problems. Do you know any relations between them and the area and volume?
 
  • #3
If you mean "parallelogram" and "rectangular prism", then the area of the parallelogram determined by the two vectors [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] is given by [itex]\vec{u}\times\vec{v}= |u||v|sin(\theta)[/itex] while the rectangular prism determined by the three vectors [itex]\vec{u}[/itex], [itex]\vec{v}[/itex], and [itex]\vec{w}[/itex] has volume [itex]\vec{u}\cdot(\vec{v}\times\vec{w})[/itex].
 
  • #4
Dick said:
PR and PQ are not orthogonal, they don't determine a rectangle. Ditto for the second problem. You should be using the cross product and the dot product to solve these problems. Do you know any relations between them and the area and volume?

You are right. We need to use Sarrus.
The b -part changes to:

b)
[tex]
Volume = Area * |\overrightarrow{PQ} x \overrightarrow{PR}|
[/tex]

[tex]
\overrightarrow{PQ} x \overrightarrow{PR} = (6, 3, 6) // by Sarrus
[/tex]

[tex]
|\overrightarrow{PQ} x \overrightarrow{PR}| = 9
[/tex]

The volume is
[tex]
Volume = 9 \sqrt{2} * 9
= 81 \sqrt {2}
[/tex]

Please, suggest any improvements.
 
  • #5
HallsofIvy said:
If you mean "parallelogram" and "rectangular prism", then the area of the parallelogram determined by the two vectors [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex] is given by [itex]\vec{u}\times\vec{v}= |u||v|sin(\theta)[/itex] while the rectangular prism determined by the three vectors [itex]\vec{u}[/itex], [itex]\vec{v}[/itex], and [itex]\vec{w}[/itex] has volume [itex]\vec{u}\cdot(\vec{v}\times\vec{w})[/itex].

I mean parallelogram.
Thank you for the correction!
 

Related to What is the volume and area determined by three points?

1. What is the volume determined by three points?

The volume determined by three points is the amount of space enclosed by the three points in a three-dimensional space. It can be calculated using the formula for the volume of a parallelepiped, which is V = |(a x b) ⋅ c|, where a, b, and c are the vectors formed by the three points.

2. How is the volume determined by three points different from the area determined by three points?

The volume determined by three points measures the space enclosed by the points in a three-dimensional space, while the area determined by three points measures the space enclosed by the points in a two-dimensional space. The volume is a three-dimensional quantity, while the area is a two-dimensional quantity.

3. What are the units of measurement for the volume and area determined by three points?

The units of measurement for the volume determined by three points are cubic units (e.g. cubic meters, cubic feet), while the units for the area determined by three points are square units (e.g. square meters, square feet). The specific units will depend on the units of measurement used for the three points.

4. Can the volume and area determined by three points be negative?

No, the volume and area determined by three points cannot be negative. These quantities represent physical measurements of space and cannot have negative values. If the result of the calculation is negative, it may indicate an error in the calculation or that the points are not arranged correctly.

5. Are there any special cases where the volume and area determined by three points cannot be calculated?

Yes, there are certain situations where the volume and area determined by three points cannot be calculated. For example, if the three points are collinear (lying on the same line), there will be no enclosed space and the volume cannot be calculated. Similarly, if the three points are coplanar (lying on the same plane), the area cannot be calculated. Additionally, the points must be non-collinear and non-coplanar for the volume and area to be uniquely determined.

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