Equations of Lines and Planes in Space

By now, we are familiar with writing equations that describe a line in two dimensions. To write an equation for a line, we must know two points on the line, or we must know the direction of the line and at least one point through which the line passes. In two dimensions, we use the concept of slope to describe the orientation, or direction, of a line. In three dimensions, we describe the direction of a line using a vector parallel to the line. In this section, we examine how to use equations to describe lines and planes in space.

Equations for a Line in Space

Let’s first explore what it means for two vectors to be parallel. Recall that parallel vectors must have the same or opposite directions. If two nonzero vectors, $\text{u}$

and $\text{v},$

are parallel, we claim there must be a scalar, $k,$

such that $\text{u}=k\text{v}.$

If $\text{u}$

and $\text{v}$

have the same direction, simply choose $k=\frac{\Vert \text{u}\Vert }{\Vert \text{v}\Vert }.$

If $\text{u}$

and $\text{v}$

have opposite directions, choose $k=-\frac{\Vert \text{u}\Vert }{\Vert \text{v}\Vert }.$

Note that the converse holds as well. If $\text{u}=k\text{v}$

for some scalar $\text{k},$

then either $\text{u}$

and $\text{v}$

have the same direction $\left(k>0\right)$

or opposite directions $\left(k<0\right),$

so $\text{u}$

and $\text{v}$

are parallel. Therefore, two nonzero vectors $\text{u}$

and $\text{v}$

are parallel if and only if $\text{u}=k\text{v}$

for some scalar $\text{k}.$

By convention, the zero vector $0$

is considered to be parallel to all vectors.

As in two dimensions, we can describe a line in space using a point on the line and the direction of the line, or a parallel vector, which we call the direction vector ([link]). Let $L$

be a line in space passing through point $P\left(x_0,y_0,z_0\right).$

Let $\text{v}=\langle a,b,c\rangle$

be a vector parallel to $L.$

Then, for any point on line $Q\left(x,y,z\right),$

we know that $\overrightarrow{PQ}$

is parallel to $\text{v}.$

Thus, as we just discussed, there is a scalar, $t,$

such that $\overrightarrow{PQ}=t\text{v},$

which gives

Using vector operations, we can rewrite [link] as

Setting $\text{r}=\langle x,y,z\rangle$

and ${\text{r}}_0=\langle x_0,y_0,z_0\rangle ,$

we now have the vector equation of a line:

Equating components, [link] shows that the following equations are simultaneously true: $x-x_0=ta,$

and $z-z_0=tc.$

If we solve each of these equations for the component variables $x,y,\text{and}z,$

we get a set of equations in which each variable is defined in terms of the parameter t and that, together, describe the line. This set of three equations forms a set of parametric equations of a line:

If we solve each of the equations for $t$

assuming $a,b,\text{and}c$

are nonzero, we get a different description of the same line:

Because each expression equals t, they all have the same value. We can set them equal to each other to create symmetric equations of a line:

We summarize the results in the following theorem.

The parametric equations of a line are not unique. Using a different parallel vector or a different point on the line leads to a different, equivalent representation. Each set of parametric equations leads to a related set of symmetric equations, so it follows that a symmetric equation of a line is not unique either.

Sometimes we don’t want the equation of a whole line, just a line segment. In this case, we limit the values of our parameter $t.$

For example, let $P\left(x_0,y_0,z_0\right)$

and $Q\left(x_1,y_1,z_1\right)$

be points on a line, and let $\text{p}=\langle x_0,y_0,z_0\rangle$

and $\text{q}=\langle x_1,y_1,z_1\rangle$

be the associated position vectors. In addition, let $\text{r}=\langle x,y,z\rangle .$

We want to find a vector equation for the line segment between $P$

and $Q.$

Using $P$

as our known point on the line, and $\overrightarrow{PQ}=\langle x_1-x_0,y_1-y_0,z_1-z_0\rangle$

as the direction vector equation, [link] gives

Using properties of vectors, then

Thus, the vector equation of the line passing through $P$

and $Q$

is

Remember that we didn’t want the equation of the whole line, just the line segment between $P$

and $Q.$

Notice that when $t=0,$

we have $r=p,$

and when $t=1,$

we have $r=q.$

Therefore, the vector equation of the line segment between $P$

and $Q$

is

Going back to [link], we can also find parametric equations for this line segment. We have

Then, the parametric equations are

Distance between a Point and a Line

We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. When building a home, for example, builders must consider “setback” requirements, when structures or fixtures have to be a certain distance from the property line. Air travel offers another example. Airlines are concerned about the distances between populated areas and proposed flight paths.

Let $L$

be a line in the plane and let $M$

be any point not on the line. Then, we define distance $d$

from $M$

to $L$

as the length of line segment $\stackrel{—}{MP},$

where $P$

is a point on $L$

such that $\stackrel{—}{MP}$

is perpendicular to $L$

([link]).

When we’re looking for the distance between a line and a point in space, [link] still applies. We still define the distance as the length of the perpendicular line segment connecting the point to the line. In space, however, there is no clear way to know which point on the line creates such a perpendicular line segment, so we select an arbitrary point on the line and use properties of vectors to calculate the distance. Therefore, let $P$

be an arbitrary point on line $L$

and let $\text{v}$

be a direction vector for $L$

([link]).

By [link], vectors $\overrightarrow{PM}$

and $\text{v}$

form two sides of a parallelogram with area $\Vert \overrightarrow{PM}\times \text{v}\Vert .$

Using a formula from geometry, the area of this parallelogram can also be calculated as the product of its base and height:

We can use this formula to find a general formula for the distance between a line in space and any point not on the line.

Relationships between Lines

Given two lines in the two-dimensional plane, the lines are equal, they are parallel but not equal, or they intersect in a single point. In three dimensions, a fourth case is possible. If two lines in space are not parallel, but do not intersect, then the lines are said to be skew lines ([link]).

To classify lines as parallel but not equal, equal, intersecting, or skew, we need to know two things: whether the direction vectors are parallel and whether the lines share a point ([link]).

Equations for a Plane

We know that a line is determined by two points. In other words, for any two distinct points, there is exactly one line that passes through those points, whether in two dimensions or three. Similarly, given any three points that do not all lie on the same line, there is a unique plane that passes through these points. Just as a line is determined by two points, a plane is determined by three.

This may be the simplest way to characterize a plane, but we can use other descriptions as well. For example, given two distinct, intersecting lines, there is exactly one plane containing both lines. A plane is also determined by a line and any point that does not lie on the line. These characterizations arise naturally from the idea that a plane is determined by three points. Perhaps the most surprising characterization of a plane is actually the most useful.

Imagine a pair of orthogonal vectors that share an initial point. Visualize grabbing one of the vectors and twisting it. As you twist, the other vector spins around and sweeps out a plane. Here, we describe that concept mathematically. Let $\text{n}=\langle a,b,c\rangle$

be a vector and $P=\left(x_0,y_0,z_0\right)$

be a point. Then the set of all points $Q=\left(x,y,z\right)$

such that $\overrightarrow{PQ}$

is orthogonal to $\text{n}$

forms a plane ([link]). We say that $\text{n}$

is a normal vector, or perpendicular to the plane. Remember, the dot product of orthogonal vectors is zero. This fact generates the vector equation of a plane: $\text{n}·\overrightarrow{PQ}=0.$

Rewriting this equation provides additional ways to describe the plane:

As described earlier in this section, any three points that do not all lie on the same line determine a plane. Given three such points, we can find an equation for the plane containing these points.

The scalar equations of a plane vary depending on the normal vector and point chosen.

Now that we can write an equation for a plane, we can use the equation to find the distance $d$

between a point $P$

and the plane. It is defined as the shortest possible distance from $P$

to a point on the plane.

Just as we find the two-dimensional distance between a point and a line by calculating the length of a line segment perpendicular to the line, we find the three-dimensional distance between a point and a plane by calculating the length of a line segment perpendicular to the plane. Let $R$

bet the point in the plane such that $\overrightarrow{RP}$

is orthogonal to the plane, and let $Q$

be an arbitrary point in the plane. Then the projection of vector $\overrightarrow{QP}$

onto the normal vector describes vector $\overrightarrow{RP},$

as shown in [link].

Parallel and Intersecting Planes

We have discussed the various possible relationships between two lines in two dimensions and three dimensions. When we describe the relationship between two planes in space, we have only two possibilities: the two distinct planes are parallel or they intersect. When two planes are parallel, their normal vectors are parallel. When two planes intersect, the intersection is a line ([link]).

We can use the equations of the two planes to find parametric equations for the line of intersection.

In addition to finding the equation of the line of intersection between two planes, we may need to find the angle formed by the intersection of two planes. For example, builders constructing a house need to know the angle where different sections of the roof meet to know whether the roof will look good and drain properly. We can use normal vectors to calculate the angle between the two planes. We can do this because the angle between the normal vectors is the same as the angle between the planes. [link] shows why this is true.

We can find the measure of the angle θ between two intersecting planes by first finding the cosine of the angle, using the following equation:

We can then use the angle to determine whether two planes are parallel or orthogonal or if they intersect at some other angle.

When we find that two planes are parallel, we may need to find the distance between them. To find this distance, we simply select a point in one of the planes. The distance from this point to the other plane is the distance between the planes.

Previously, we introduced the formula for calculating this distance in [link]:

where $Q$

is a point on the plane, $P$

is a point not on the plane, and $\text{n}$

is the normal vector that passes through point $Q.$

Consider the distance from point $\left(x_0,y_0,z_0\right)$

to plane $ax+by+cz+k=0.$

Let $\left(x_1,y_1,z_1\right)$

be any point in the plane. Substituting into the formula yields

We state this result formally in the following theorem.

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 $\text{v}=\langle a,b,c\rangle$

  passing through point

  $P=\left(x_0,y_0,z_0\right)$

  is

  $\text{r}={\text{r}}_0+t\text{v},$

  where

  ${\text{r}}_0=\langle x_0,y_0,z_0\rangle$

  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

  $\text{n}=\langle a,b,c\rangle$

  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

  ---

  $D=\Vert {\text{proj}}_{\text{n}}\overrightarrow{QP}\Vert =\backslash |{\text{comp}}_{\text{n}}\overrightarrow{QP}\backslash |=\frac{\backslash |\overrightarrow{QP}·\text{n}\backslash |}{\Vert \text{n}\Vert }.$
• 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:

  $\text{cos}\theta =\frac{\backslash |{\text{n}}_1·{\text{n}}_2\backslash |}{\Vert {\text{n}}_1\Vert \Vert {\text{n}}_2\Vert },$

  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

  ---

  $D=\frac{\backslash |a\left(x_0-x_1\right)+b\left(y_0-y_1\right)+c\left(z_0-z_1\right)\backslash |}{\sqrt{a^2+b^2+c^2}}=\frac{\backslash |a{x}_0+b{y}_0+c{z}_0+d\backslash |}{\sqrt{a^2+b^2+c^2}}.$

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

  ---

  $d=\Vert {\text{proj}}_{\text{n}}\overrightarrow{QP}\Vert =\backslash |{\text{comp}}_{\text{n}}\overrightarrow{QP}\backslash |=\frac{\backslash |\overrightarrow{QP}·\text{n}\backslash |}{\Vert \text{n}\Vert }$