Altitudes of tetrahedron intersect, coordinate vectors

**acevipa** · June 3rd 2010, 03:37 AM

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

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

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:

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

Not too sure if that's right

**earboth** · June 3rd 2010, 04:07 AM

Originally Posted by acevipa

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

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

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:

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

Not too sure if that's right

The equations of the lines containing the altitudes are correct.

BTW: The two lines don't intersect.

**acevipa** · June 3rd 2010, 04:13 AM

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?

**earboth** · June 3rd 2010, 04:49 AM

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.

**acevipa** · June 3rd 2010, 05:11 AM

Originally Posted by earboth

Exactly! Nothing else.

Thanks mate

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