# Thread: Altitudes of tetrahedron intersect, coordinate vectors

1. ## Altitudes of tetrahedron intersect, coordinate vectors

A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being:

$\displaystyle \mathbf{a}=\begin{pmatrix}0\\1\\2\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}-1\\4\\1\end{pmatrix},$ $\displaystyle \mathbf{c}= \begin{pmatrix}1\\0\\3\end{pmatrix},\ and\ \mathbf{d}=\begin{pmatrix}-3\\1\\2\end{pmatrix}$

Find parametric vector equations for the two altitudes of the tetrahedron which pass through the vertices A and B, and determine whether the two altitudes intersect or not.

Altitude is a line through the vertex and perpendicular to the opposite face.

For my altitudes, I managed to get:

$\displaystyle \mathbf{x}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\la mbda\begin{pmatrix}-2\\6\\14\end{pmatrix} and\ \mathbf{x}=\begin{pmatrix}-1\\4\\1\end{pmatrix}+\lambda\begin{pmatrix}0\\-3\\3\end{pmatrix}$

Not too sure if that's right

2. Originally Posted by acevipa
A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being:

$\displaystyle \mathbf{a}=\begin{pmatrix}0\\1\\2\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}-1\\4\\1\end{pmatrix},$ $\displaystyle \mathbf{c}= \begin{pmatrix}1\\0\\3\end{pmatrix},\ and\ \mathbf{d}=\begin{pmatrix}-3\\1\\2\end{pmatrix}$

Find parametric vector equations for the two altitudes of the tetrahedron which pass through the vertices A and B, and determine whether the two altitudes intersect or not.

Altitude is a line through the vertex and perpendicular to the opposite face.

For my altitudes, I managed to get:

$\displaystyle \mathbf{x}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\la mbda\begin{pmatrix}-2\\6\\14\end{pmatrix} and\ \mathbf{x}=\begin{pmatrix}-1\\4\\1\end{pmatrix}+\lambda\begin{pmatrix}0\\-3\\3\end{pmatrix}$

Not too sure if that's right
The equations of the lines containing the altitudes are correct.

BTW: The two lines don't intersect.

3. Originally Posted by earboth
The equations of the lines containing the altitudes are correct.
Ok great, so when it asks would they intersect, would you just equate the 2 lines and so the answer would be, no, they do not intersect?

4. Originally Posted by acevipa
Ok great, so when it asks would they intersect, would you just equate the 2 lines and so the answer would be, no, they do not intersect?
Exactly! Nothing else.

5. Originally Posted by earboth
Exactly! Nothing else.
Thanks mate