2.5 Equations of lines and planes in space  (Page 8/19)

Key concepts

• In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector $\text{v}=⟨a,b,c⟩$ passing through point $P=\left(x_0,y_0,z_0\right)$ is $\text{r}={\text{r}}_0+t\text{v},$ where ${\text{r}}_0=⟨x_0,y_0,z_0⟩$ is the position vector of point $P.$ This equation can be rewritten to form the parametric equations of the line: $x=x_0+ta,$ $y=y_0+tb,$ and $z=z_0+tc.$ The line can also be described with the symmetric equations $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}.$
• Let $L$ be a line in space passing through point $P$ with direction vector $\text{v}.$ If $Q$ is any point not on $L,$ then the distance from $Q$ to $L$ is $d=\frac{\Vert \overrightarrow{PQ}\times \text{v}\Vert }{\Vert \text{v}\Vert }.$
• In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
• Given a point $P$ and vector $\text{n},$ the set of all points $Q$ satisfying equation $\text{n}·\overrightarrow{PQ}=0$ forms a plane. Equation $\text{n}·\overrightarrow{PQ}=0$ is known as the vector equation of a plane.
• The scalar equation of a plane containing point $P=\left(x_0,y_0,z_0\right)$ with normal vector $\text{n}=⟨a,b,c⟩$ is $a\left(x-x_0\right)+b\left(y-y_0\right)+c\left(z-z_0\right)=0.$ This equation can be expressed as $ax+by+cz+d=0,$ where $d=\text{−}a{x}_0-b{y}_0-c{z}_0.$ This form of the equation is sometimes called the general form of the equation of a plane .
• Suppose a plane with normal vector n passes through point $Q.$ The distance $D$ from the plane to point $P$ not in the plane is given by
  
  $D=\Vert {\text{proj}}_{\text{n}}\overrightarrow{QP}\Vert =\left|{\text{comp}}_{\text{n}}\overrightarrow{QP}\right|=\frac{\left|\overrightarrow{QP}·\text{n}\right|}{\Vert \text{n}\Vert }.$
• The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
• The measure of the angle $\theta$ between two intersecting planes can be found using the equation: $\text{cos}\theta =\frac{\left|{\text{n}}_1·{\text{n}}_2\right|}{\Vert {\text{n}}_1\Vert \Vert {\text{n}}_2\Vert },$ where ${\text{n}}_1$ and ${\text{n}}_2$ are normal vectors to the planes.
• The distance $D$ from point $\left(x_0,y_0,z_0\right)$ to plane $ax+by+cz+d=0$ is given by
  
  $D=\frac{\left|a\left(x_0-x_1\right)+b\left(y_0-y_1\right)+c\left(z_0-z_1\right)\right|}{\sqrt{a^2+b^2+c^2}}=\frac{\left|a{x}_0+b{y}_0+c{z}_0+d\right|}{\sqrt{a^2+b^2+c^2}}.$

Key equations

• Vector Equation of a Line
  $\text{r}={\text{r}}_0+t\text{v}$
• Parametric Equations of a Line
  $x=x_0+ta,$ $y=y_0+tb,$ and $z=z_0+tc$
• Symmetric Equations of a Line
  $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$
• Vector Equation of a Plane
  $\text{n}·\overrightarrow{PQ}=0$
• Scalar Equation of a Plane
  $a\left(x-x_0\right)+b\left(y-y_0\right)+c\left(z-z_0\right)=0$
• Distance between a Plane and a Point
  $d=\Vert {\text{proj}}_{\text{n}}\overrightarrow{QP}\Vert =\left|{\text{comp}}_{\text{n}}\overrightarrow{QP}\right|=\frac{\left|\overrightarrow{QP}·\text{n}\right|}{\Vert \text{n}\Vert }$

In the following exercises, points $P$ and $Q$ are given. Let $L$ be the line passing through points $P$ and $Q.$

1. Find the vector equation of line $L.$
2. Find parametric equations of line $L.$
3. Find symmetric equations of line $L.$
4. Find parametric equations of the line segment determined by $P$ and $Q.$

For the following exercises, point $P$ and vector $\text{v}$ are given. Let $L$ be the line passing through point $P$ with direction $\text{v}.$

1. Find parametric equations of line $L.$
2. Find symmetric equations of line $L.$
3. Find the intersection of the line with the xy -plane.

For the following exercises, line $L$ is given.

1. Find point $P$ that belongs to the line and direction vector $\text{v}$ of the line. Express $\text{v}$ in component form.
2. Find the distance from the origin to line $L.$