LEC9

• Vector equation of a line.
• Parametric equation of a line.
• Vector equation of a plane.

Vector Equation of a Line

For a line, we basically need two things:

• A point through which it passes.
• A vector $\vec{d}$ in the direction of the line.

In $R^2$

$$\begin{array}{r}y=mx+c\\ x_2=m{x}_1+c\end{array}$$

In the traditional form, $m$ is similar, though not the same, to the direction vector.

In $R^n$

$$\vec{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ .\\ .\\ x_n\end{array}\right]=\vec{p}+t\vec{d}$$

• $\vec{p}$ is the position vector of the point through which our line passes.
• $\vec{d}$ is the direction vector.
• $t$ is just a parameter used to figure out the values at different points.

Parametric Equation of a Line

• The vector equation is more general, representing all possible dimensions.
• If we break it down:

$$\begin{array}{r}{x}_1=p_1+t{d}_1\\ x_2=p_2+t{d}_2\\ .\\ .\\ .\\ x_n=p_n+t{d}_n\end{array}$$

Vector Equation of a Plane

• Similar to a line, we have a point that lies on the plane.
• But instead of just one direction vector, like in line, we need two direction vectors to represent a plane.

$$\vec{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ .\\ .\\ x_n\end{array}\right]=\vec{p}+s\vec{u}+t\vec{v}$$

• $u,v$ are non-zero and non-parallel (if parallel then they’re the same vector but scaled!, hence can’t represent a place)