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Mathematics LibreTexts

1.6: Lines and Planes

Lines

Our goal is to come up with the equation of a line given a vector v parallel to the line and a point (a,b,c) on the line. The figure (shown in 2D for simplicity) shows that if P is a point on the line then

\[ \langle x,y \rangle = P + tv\]

for some number \(t\).  

The picture is the same for 3D.  The formula is given below.

Note: Parametric Equations of a Line

The parametric equations for the line through the point \((a,b,c)\) and parallel to the vector v are

\[<x,y,z> = <a,b,c> + t\textbf{v}.\]

Example 1

Find the parametric equations of the line that passes through the point \((1, 2, 3)\) and is parallel to the vector <4, -2, 1>.

Solution

We write:

\[<x, y, z> = <1, 2, 3> + t <4, -2, 1> = <1 + 4t, 2 - 2t, 3 + t>\]

or

\[ x(t) = 1 + 4t,\]

\[y(t) = 2 - 2t,\]

\[z(t) = 3 + t.\]

Exercise 1

Find the parametric equations of the line through the two points \((2,1,7)\) and \((1,3,5)\).

Hint: a vector parallel to the line has tail at \((2,1,7)\) and head at \((1,3,5)\).

Planes

If S is a plane then a vector n is normal (perpendicular) to the plane if it is orthogonal to every vector that lies on the plane.  Suppose that n is a normal vector to a plane and \((a,b,c)\) is a point on the plane.  Let \((x,y,z)\) be a general point on the plane, then

\[ <x - a, y - b, z - c>\]

is parallel to the plane, hence

\[\vec{n} \cdot <x - a, y - b, z - c> = 0.\]

This defines the equation of the plane.

Example 2

Find the equation of the plane that contains the point \((2,1,0)\) and has normal vector \(<1,2,3>\).

Solution

We have

\[<1,2,3> \cdot <x - 2,y - 1,z - 0>  =  0\]

so that

\[1(x - 2) + 2(y - 1) + 3z  =  0\]

or

\[ x + 2y + 3z  =  4.\]

Example 3

Find the equation of the plane through the points 

  • \(P  =  (0,0,1)\)
  • \(Q  =  (2,1,0)\)
  • \(R  =  (1,1,1)\)

Solution

Let 

\[\textbf{v} = Q-P = <2,1,-1>  \]

and

\[\textbf{w} = R-P = <1,1,0>  \]

then to find a vector normal to the plane, we find the cross product of \(v\) and \(w\):

\[v \times w = \begin{vmatrix} \hat{\textbf{i}} &  \hat{\textbf{i}} & \hat{\textbf{i}}  \\ 2 & 1 &-1 \\ 1 &1 0   \end{vmatrix} =  \hat{\textbf{i}} -  \hat{\textbf{j}} +  \hat{\textbf{k}} \]

or 

\[<1, -1, 1>.\]

We can now use the formula:

\[<1, -1, 1> \cdot <x, y, z - 1>  =  0\]

or

\[x - y + z - 1  =  0\]

or

\[x - y + z  =  1\]

Distance Between a Point and a Plane

Let P be a point and Q be a point on a plane with normal vector n, then the distance between P and the plane is given by

Note:  Distance Between a Point \(P\) and a Plane With Normal Vector n

Let \(Q\) be a point on the plane with normal vector \(\vec{n}\).  The the distance from the point \(P\) to this plane is given by

\[ Proj_nPQ  =  \dfrac{ ||PQ \cdot \vec{n}|| }{ || \vec{n}|| }.\]

Example 4

Find the distance between the point \((1,2,3)\) and the plane

\[2x - y - 2z = 5.\]

Solution

The normal vector can be read off from the equation as

\[\vec{n}  =  <2, -1, -2>.\]

 

Now find a convenient point on the plane such as \(Q  =  (0, -5, 0)\).  We have 

\[Q  =  <-1, -7, -3>\]

and 

\[ \vec{n} \cdot PQ  =  -2 + 7 + 6  =  11.\]

We find the magnitude of n by taking the square root of the sum of the squares.  The sum is

\[4 + 1 + 4  =  9\]

so 

\[|| \vec{n} ||  =  3.\]

Hence the distance from the point to the plane is \(\frac{11}{3}\).

The Angle Between 2 Planes

The angle between two planes is given by the angle between the normal vectors.

Example

Find the angle between the two planes

\[ 3x - 2y + 5z  =  1\]

and

\[4x + 2y - z  =  4.\]

We have the two normal vectors are

\[ \vec{n} = <3,-2,5>\]

and 

\[\vec{m} = <4,2,-1>.\]

We have

\[\vec{n} \cdot \vec{ m}  =  3\]

\[ || \vec{n} || = \sqrt{38}\]

\[ || \vec{m} || = \sqrt{21}\]

hence the angle is

\[ \cos^{-1} \left(\dfrac{3}{\sqrt{38}\sqrt{21}} \right) = 1.46 \, rad.\]

Contributors

  • Integrated by Justin Marshall.