Lines and Planes in 3D - Kennesaw State Universityksuweb.kennesaw.edu/~plaval/math2203/space_linesplanes_slides.pdfLines and Planes in 3D Philippe B. Laval KSU January 28, 2013 Philippe

Lines and Planes in 3D

Philippe B. Laval

KSU

January 28, 2013

Philippe B. Laval (KSU) Lines and Planes in 3D January 28, 2013 1 / 20

Introduction

Recall that given a point P = (a, b, c), one can draw a vector from theorigin to P. Such a vector is called the position vector of the point Pand its coordinates are 〈a, b, c〉, the same as P. Position vectors areusually denoted −→r .In this section, we derive the equations of lines and planes in 3-D . We doso by finding the conditions a point P = (x , y , z) or its correspondingposition vector −→r = 〈x , y , z〉 must satisfy in order to belong to the objectbeing studied (line or plane).


Lines in 3D: Statement of the Problem

In 3-D, like in 2-D, a line is uniquely determined when one point on theline and the direction of the line are given. In this section, we assume weare given a point P0 = (x0, y0, z0) on the line and a direction vector−→v = 〈a, b, c〉. Our goal is to determine the equation of the line L whichgoes through P0 and is parallel to

−→v .

Definitiona, b, and c are called the direction numbers of the line L.

Let P (x , y , z) be an arbitrary point on L. We wish to find the conditionsP must satisfy to be on the line L.


Lines in 3D

Figure: Line through P0 parallel to−→vPhilippe B. Laval (KSU) Lines and Planes in 3D January 28, 2013 4 / 20

Lines in 3D: Vector Equation

Looking at the figure on the previous slide, we see that a necessary andsuffi cient condition for the point P to be on the line L is that

−−→P0P be

parallel to −→v .This gives us

−→r = −→r0 + t−→v

DefinitionThe above equation is known as the vector equation of the line L. Thescalar t used in the equation is called a parameter.

The parameter t can be any real number. As it varies, the point P movesalong the line. When t = 0, P is the same as P0. When t > 0, P is awayfrom P0 in the direction of

−→v and when t < 0, P is away from P0 in thedirection opposite −→v . The larger t is (in absolute value), the further awayP is from P0.


Lines in 3D: Parametric Equation

If we switch to coordinates, the vector equation becomes〈x , y , z〉 = 〈x0, y0, z0〉+ t 〈a, b, c〉. Two vectors are equal when theircorresponding coordinates are equal. Thus, we obtain

x = x0 + aty = y0 + btz = z0 + ct

DefinitionThe above equation is known as the parametric equation of the line L.


Lines in 3D: Symmetric Form

If we solve for t in the parametric equation, assuming that a 6= 0, b 6= 0,and c 6= 0 we obtain

x − x0a

=y − y0b

=z − z0c

DefinitionThe above equations are known as the symmetric equations of the line L.

If a = 0, we only solve for t in the last two parametric equations to get

x = x0y − y0b

=z − z0c

We would obtain a similar result if one of the other direction numbers is 0.


Lines in 3D: Examples

ExampleFind the parametric and symmetric equations of the line throughP (−1, 4, 2) in the direction of −→v = 〈1, 2, 3〉

ExampleFind the parametric and symmetric equations of the line throughP1 (1, 2, 3) and P2 (2, 4, 1).


Lines in 3D: Line Segment

We can also derive a formula for the equation of a line given two points onthe line: P0 = (x0, yo , z0) and P1 = (x1, y1, z1). Using P0 for the point and−−−→P0P1 for the direction vector, we get:

x = x0 + t (x1 − x0)y = y0 + t (y1 − y0)z = z0 + t (z1 − z0)

Factoring differently givesx = (1− t) x0 + tx1y = (1− t) y0 + ty1z = (1− t) z0 + tz1

When t = 0, we are at the point P0 and when t = 1, we are at the pointP1. So, if t is allowed to take on any real value, then this equation willdescribe the whole line. On the other hand, if we restrict t to [0, 1], thenthis equation describes the portion of the line between P0 and P1 which iscalled the line segment from P0 to P1.


Lines in 3D: Line Segment

ExampleFind the equation of the line segment from P0 = (1, 2, 3) to P1 (2, 4, 1).


Lines in 3D: Intersecting, Parallel and Skew Lines

Recall that in 2-D two lines were either parallel or intersected. In 3-Dit is also possible for two lines to not be parallel and to not intersect.Such lines are called skew lines.

Two lines are parallel if their direction vectors are parallel.

If two lines are not parallel, we can find if they intersect if there existsvalues of the parameters in their equations which produce the samepoint. More specifically, if the first line has equation −→r0 + t−→v and thesecond line has equation

−→R0 + s

−→u then they will intersect if thereexists a value for t and s such that −→r0 + t−→v =

−→R0 + s

−→u .Also, when two lines intersect, we can find the angle between them byfinding the smallest angle between their direction vectors (using thedot product).

Finally, two lines are perpendicular if their direction vectors areperpendicular.



Recall that in 2-D two lines were either parallel or intersected. In 3-Dit is also possible for two lines to not be parallel and to not intersect.Such lines are called skew lines.Two lines are parallel if their direction vectors are parallel.


−→R0 + s


−→R0 + s







−→R0 + s


−→R0 + s

−→u .

Also, when two lines intersect, we can find the angle between them byfinding the smallest angle between their direction vectors (using thedot product).






−→R0 + s


−→R0 + s







−→R0 + s


−→R0 + s





ExampleLet L1 be the line through (1,−6, 2) with direction vector 〈1, 2, 1〉 and L2be the line through (0, 4, 1) with direction vector 〈2, 1, 2〉. Determine ifthe lines are parallel, if they intersect or if they are skew. If they intersect,find the point at which they intersect.

ExampleFind the angle between the two previous lines.

ExampleFind the points at which L1 in the example above intersects with thecoordinate planes.


Lines in 3D: Summary for Lines

1 Be able to find the equation of a line given a point and a direction orgiven two points.

2 Be able to tell if two lines are parallel, intersect or are skewed.3 Be able to find the angle between two lines which intersect.4 Be able to find the points at which a line intersect with thecoordinate planes.


Planes in 3D: Statement of the Problem

A plane is uniquely determined given a point on the plane and a vectorperpendicular to the plane. Such a vector is said to be normal to theplane. To help visualize this, consider the figure below.

Figure: Plane determined by a point and its normalPhilippe B. Laval (KSU) Lines and Planes in 3D January 28, 2013 14 / 20

Planes in 3D: Equation Given a Point and a VectorPerpendicular

Given a point P0 = (x0, y0, z0) and a normal ~n = 〈a, b, c〉 to a plane, apoint P = (x , y , z) will be on the plane if

−−→P0P is perpendicular to ~n that is

−→n · −−→P0P = 0This is known as the vector equation of a plane. Switching tocoordinates, we get

ax + by + cz + d = 0

where d = −ax0 − by0 − cz0. This is known as the scalar equation of aplane. It is also the equation of a plane in implicit form.

FactNote that when we know the scalar equation of a plane, we automaticallyknow its normal; it is given by the coeffi cients of x, y , and z.

FactA plane in 3D is the analogous of a line in 2D.


Planes in 3D: Examples

ExampleWhat is the normal to the plane 3x + 2y − z = 10?

ExampleWhat is the scalar equation of a plane through (1, 2, 3) with normal〈2, 1, 4〉?

ExampleFind the equation of the plane through the points P1 (0, 1, 1), P2 (1, 0, 1)and P3 (1,−3,−1).


Planes in 3D: Parallel and Intersecting Planes

Two planes are parallel if and only if their normals are parallel.

If two planes p1 with normal−→n1 and p2 with normal −→n2 are not

parallel, then the angle θ between them is defined to be the smallestangle between their normals that is the angle with the non-negativecosine. In other words,

θ = cos−1∣∣−→n1 · −→n2∣∣∥∥−→n1∥∥∥∥−→n2∥∥ (1)


Planes in 3D: Parallel and Intersecting Planes

Two planes are parallel if and only if their normals are parallel.

If two planes p1 with normal−→n1 and p2 with normal −→n2 are not

parallel, then the angle θ between them is defined to be the smallestangle between their normals that is the angle with the non-negativecosine. In other words,

θ = cos−1∣∣−→n1 · −→n2∣∣∥∥−→n1∥∥∥∥−→n2∥∥ (1)


Planes in 3D: Examples

ExampleFind the angle between the planes p1 : x + y + z − 1 = 0 andp2 : x − 2y + 3z − 1 = 0. Find their intersection.


Planes in 3D: Summary for Planes

In addition, using the material studied so far, you should be able to do thefollowing:

1 Be able to find the equation of a plane given a point on the plane anda normal to the plane.

2 Be able to find the equation of a plane given three points on theplane.

3 Be able to find the equation of a plane through a point and parallel toa given plane.

4 Be able to find the equation of a plane through a point and a line notcontaining the point.

5 Be able to tell if two planes are parallel, perpendicular.6 Be able to find the angle between two planes.7 Be able to find the traces of a plane.8 Be able to find the intersection of two planes.


Exercises

See the problems at the end of my notes on equations of lines and planesin 3D.


http://science.kennesaw.edu/~plaval/math2203/space_linesplanes.pdf

http://science.kennesaw.edu/~plaval/math2203/space_linesplanes.pdf

Documents

Lines and Planes in 3D - Kennesaw State Universityksuweb.kennesaw.edu/~plaval/math2203/space_linesplanes_slides.pdfLines and Planes in 3D Philippe B. Laval KSU January 28, 2013 Philippe