# What is the volume and area determined by three points?

1. Jan 19, 2009

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

Last edited: Jan 19, 2009
2. Jan 19, 2009

### 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?

3. Jan 19, 2009

### HallsofIvy

Staff Emeritus
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 $\vec{u}\cdot(\vec{v}\times\vec{w})$.

4. Jan 19, 2009

### soopo

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