# Find the vector equation of the line perpendicular to the vectors 3bb(ulhati)-4bb(ul hatj)+bb(ul hatk) and 2bb(ul hatj)+3bb(ul hatk)?

Mar 20, 2017

$\boldsymbol{\vec{r}} = \left(\begin{matrix}2 \\ 1 \\ - 1\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}- 14 \\ - 9 \\ 6\end{matrix}\right)$

#### Explanation:

First consider the vectors $3 \boldsymbol{\underline{\hat{i}}} - 4 \boldsymbol{\underline{\hat{j}}} + \boldsymbol{\underline{\hat{k}}}$ and $2 \boldsymbol{\underline{\hat{j}}} + 3 \boldsymbol{\underline{\hat{k}}}$. A vector, $\boldsymbol{\vec{n}}$, perpendicular to these vectors is found by forming the cross product:

$\boldsymbol{\vec{n}} = \left(\begin{matrix}3 \\ - 4 \\ 1\end{matrix}\right) \times \left(\begin{matrix}0 \\ 2 \\ 3\end{matrix}\right)$

$\setminus \setminus \setminus \setminus = | \left(\boldsymbol{\underline{\hat{i}}} , \boldsymbol{\underline{\hat{j}}} , \boldsymbol{\underline{\hat{k}}}\right) , \left(3 , - 4 , 1\right) , \left(0 , 2 , 3\right) |$

$\setminus \setminus \setminus \setminus = | \left(- 4 , 1\right) , \left(2 , 3\right) | \boldsymbol{\underline{\hat{i}}} - | \left(3 , 1\right) , \left(0 , 3\right) | \boldsymbol{\underline{\hat{j}}} + | \left(3 , - 4\right) , \left(0 , 2\right) | \boldsymbol{\underline{\hat{k}}}$

$\setminus \setminus \setminus \setminus = \left(- 12 - 2\right) \boldsymbol{\underline{\hat{i}}} - \left(9 - 0\right) \boldsymbol{\underline{\hat{j}}} + \left(6 - 0\right) \boldsymbol{\underline{\hat{k}}}$
$\setminus \setminus \setminus \setminus = - 14 \boldsymbol{\underline{\hat{i}}} - 9 \boldsymbol{\underline{\hat{j}}} + 6 \boldsymbol{\underline{\hat{k}}}$

So now that we have the vector that the line is in the direction of, so the line equation is given by:

$\boldsymbol{\vec{r}} = \boldsymbol{\vec{O G}} + l a m \mathrm{da} \boldsymbol{\vec{r}}$
$\setminus \setminus \setminus \setminus = \left(\begin{matrix}2 \\ 1 \\ - 1\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}- 14 \\ - 9 \\ 6\end{matrix}\right)$

We can confirm this with a 3D diagram: 