# Convert $\begin{pmatrix}7\\2\\3\end{pmatrix}=\lambda\begin{pmatrix}0\\5\\2\end{pmatrix}$ to cartesian form

Convert \begin{pmatrix}7\\2\\3\end{pmatrix}=\lambda\begin{pmatrix}0\\5\\2\end{pmatrix} to cartesian form

• x=7+0\lambda \Rightarrow x=7
• y=2+5\lambda \Rightarrow \lambda=\frac{y-2}{5}
• z=3+2\lambda \Rightarrow \lambda=\\frac{z-3}{2}
• Answer: x=7, \frac{y-2}{5}=\frac{z-3}{2}

Find the cartesian equation of the line between 2,3,-2) and (5,8,5)

• Direction vector \vec d = \begin{pmatrix}5-2\\8-3\\5-(-2)\end{pmatrix}=\begin{pmatrix}3\\5\\7\end{pmatrix}
• Vector equation: \vec r = \begin{pmatrix}2\\3\\-2\end{pmatrix} + \lambda \begin{pmatrix}3\\5\\7\end{pmatrix}
• x=2+3\lambda \Rightarrow \lambda=\frac{x-2}{3}
• y=3+5\lambda \Rightarrow \lambda=\frac{y-3}{5}
• z=-2+7\lambda \Rightarrow \lambda=\frac{z+2}{7}
• Answer: \frac{x-2}{3}=\frac{y-3}{5}=\frac{z+2}{7}