- #1
GunnaSix
- 35
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Homework Statement
Let [itex]\vec{a},\vec{b},\vec{c}[/itex] be three constant vectors drawn from the origin to the points [itex]A,B,C[/itex]. What is the distance from the origin to the plane defined by the points [itex]A,B,C[/itex]? What is the area of the triangle [itex]ABC[/itex]?
Homework Equations
The Attempt at a Solution
Starting with the first part:
The distance is the length of a perpendicular vector from the origin to the plane. If that vector is [itex]\vec{r}[/itex], then
[tex]\vec{r} \cdot (\vec{a} - \vec{b}) = \vec{r} \cdot (\vec{b} - \vec{c}) = \vec{r} \cdot (\vec{c} - \vec{a}) = \vec{r} \cdot (\vec{a} - \vec{r}) = 0\\ \vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = r^2[/tex]
by perpendicularity.
I can also get
[tex]\vec{r} \times [(\vec{a} - \vec{b}) \times (\vec{b} - \vec{c})] = 0[/tex]
by a similar argument, but I don't think it really helps.
I'm stuck here. Is there another relationship that I'm missing?