What is the volume and area determined by three points?

[soopo]

1. The problem statement, all variables and given/known data
Let P = (2, 2, 0), Q = (0, 4, 1) and R = (-1, 2, 3) in the space \Re^{3}.

a) Determine the area of the rectangle determined by vectors \overrightarrow{PQ} and \overrightarrow{PR}.
b) Determine the volume of the tetrahedral determined by vectors \overrightarrow{PQ} and \overrightarrow{PR}, and the origin O, OPQR.

3. The attempt at a solution
a)

|\overrightarrow{PQ}| = \sqrt {4 + 4 +1} = 3



|\overrightarrow{PR}| = \sqrt {9 + 0 + 9} = 3 \sqrt {2}



Area = |\overrightarrow{PQ}| * |\overrightarrow{PR}|
= 9 \sqrt {2}


b)

Volume = Area * |\overrightarrow{RO}|
|\overrightarrow{RO}| = \sqrt {1 + 4 + 9}
= \sqrt {14}


The volume is

Volume = 9 \sqrt{2} * \sqrt {14}
= 18 \sqrt {7}


Please, comment any mistakes.

[Dick]

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?

[HallsofIvy]

If you mean "parallelogram" and "rectangular prism", then the area of the parallelogram determined by the two vectors \vec{u} and \vec{v} is given by \vec{u}\times\vec{v}= |u||v|sin(\theta) while the rectangular prism determined by the three vectors \vec{u}, \vec{v}, and \vec{w} has volume \vec{u}\cdot(\vec{v}\times\vec{w}).

[soopo]

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)

Volume = Area * |\overrightarrow{PQ} x \overrightarrow{PR}|



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



|\overrightarrow{PQ} x \overrightarrow{PR}| = 9


The volume is

Volume = 9 \sqrt{2} * 9
= 81 \sqrt {2}


Please, suggest any improvements.

[soopo]

If you mean "parallelogram" and "rectangular prism", then the area of the parallelogram determined by the two vectors \vec{u} and \vec{v} is given by \vec{u}\times\vec{v}= |u||v|sin(\theta) while the rectangular prism determined by the three vectors \vec{u}, \vec{v}, and \vec{w} has volume \vec{u}\cdot(\vec{v}\times\vec{w}).

I mean parallelogram.
Thank you for the correction!