A line is determined by a point with position vector $\mathbf a$ and a nonzero direction vector \(

Question

A line is determined by a point with position vector $\mathbf a$ and a nonzero direction vector $\mathbf d$.

Vector and parametric equations

\[\mathbf r=\mathbf a+\lambda\mathbf d,\qquad \begin{cases}x=a_1+\lambda d_1,\\y=a_2+\lambda d_2,\\z=a_3+\lambda d_3.\end{cases}\]

Cartesian form

\[\frac{x-a_1}{d_1}=\frac{y-a_2}{d_2}=\frac{z-a_3}{d_3}\]

The symmetric Cartesian form is written only for nonzero $d_1,d_2,d_3$. If, for example, $d_1=0$, write $x=a_1$ and retain the remaining parametric or symmetric relations. Through points $A,B$, use $\mathbf d=\mathbf b-\mathbf a$.

Accepted Answer

$\mathbf r=\begin{pmatrix}2\-5\5\end{pmatrix}+\lambda\begin{pmatrix}1\1\-1\end{pmatrix},\ \lambda\in\mathbb R$ $x-2=y+5=5-z$

Equations of Lines

Equations of Lines

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