\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \)
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Arc Length
Surface Area
Polar Coordinates
Slope & Tangent Lines
Arc Length
Surface Area
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Practice Exams
17calculus on YouTube
More Math Help
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Instructor or Coach?
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vectors > planes in 3-space

ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

free ideas to save on books - bags - supplies

learning and study techniques

Join Amazon Student - FREE Two-Day Shipping for College Students

The Humongous Book of Calculus Problems

Planes in 3-Space Using Vectors

Planes in 3-space are defined by one of the following

3 points that are not collinear,

one point and 2 non-parallel vectors, or

one point and a normal vector

For calculations, it is most convenient to have one point and a normal vector. You can get these from the other two as follows.

If you have one point and 2 non-parallel vectors that lie in the plane, you can take the cross product of the two vectors to get a vector that is perpendicular to the original 2 vectors and perpendicular to the plane (a normal vector). See the practice problems below for examples.

If you have 3 points that are not collinear, we use these points to get two vectors. Then, by choosing one of the points (any of the three will work), we have the previous case of one point and 2 non-parallel vectors. We then use that procedure to get a normal vector. See the practice problems below for examples.

There may be times that you are given a point and a normal vector and you need either two vectors in the plane or three points in the plane. See the practice problems below for examples on how to extract that information.

You can determine an incredible amount of information about a plane and it's relationship to other planes just knowing a point and a normal vector. Examples include the angle between two planes including whether two planes are perpendicular or parallel, the distance between a point or line and a plane and other equations to define a plane. Most of these calculations are covered on this page. For now, we continue our discussion on how to define a plane.

Here is a good video clip explaining this idea again.

PatrickJMT - planes in space

General and Standard Forms

The (simplified) general form of the equation of a plane is \( Ax + By + Cz + D = 0 \). The cool thing about this form is that the vector formed from the coefficients of the \(x,y,z\) components, i.e. \( \langle A,B,C \rangle \) is a normal vector to the plane. To get one point on the plane, we just choose values for \(x\) and \(y\) and solve for \(z\).

Alternatively, if we are given one point \((x_0, y_0, z_0)\) and a normal vector \(\langle A,B,C \rangle\), we can derive the general form of the equation of the plane using the standard equation \(A(x-x_0)+B(y-y_0)+C(z-z_0)=0\). [proof] By multiplying out, we can calculate d and end up with an equation in the form \( Ax + By + Cz + D = 0 \). In the simplified general form, the constants \( A,B,C,D \) are all integers and have no common factor.

Proof of the Standard Equation of a Plane

Proof of the Standard Equation of a Plane

Theorem: Standard Equation of a Plane

Given a point \( P = ( x_0, y_0, z_0 ) \) in a plane and a normal vector \( \vec{n} = \langle A,B,C \rangle \) to the plane, the standard equation of the plane is
\[ A(x-x_0) + B(y-y_0) + C(z-z_0) = 0 \]

Proof - - Let the point \( Q = (x,y,z) \) be any point in the plane. We can form a vector in the plane \( \overrightarrow{PQ} = \langle x-x_0, y-y_0, z-z_0 \rangle \).
Since the normal vector is orthogonal to vector \( \overrightarrow{PQ} \) in the plane, we know that the dot product is zero. So we can write

\( \begin{array}{rcl} \vec{n} \cdot \overrightarrow{PQ} & = & 0 \\ \langle A,B,C \rangle \cdot \langle x-x_0, y-y_0, z-z_0 \rangle & = & 0 \\ A(x-x_0) + B(y-y_0) + C(z-z_0) & = & 0 ~~~ \text{[qed]} \end{array} \)

Parametric Equations

We can define a plane parametrically using two parameters. For this discussion, we will use the greek letters \( \lambda \) and \( \mu \).

Given one point \( \langle x_0, y_0, z_0 \rangle \) and two non-parallel vectors in the plane, we can get a set of parametric equations similar to how we did above for lines. Let's call the two vectors \(\vec{a} = \langle a_x,a_y,a_z \rangle \) and \(\vec{b} = \langle b_x,b_y,b_z \rangle\).

Using the two parameters \( \lambda \) and \( \mu \), any point on the plane can be written as \( \langle x,y,z \rangle = \langle x_0, y_0, z_0 \rangle + \lambda \vec{a} + \mu \vec{b} \). Equating each component, we get the three parametric equations

\( x = x_0 + \lambda a_x + \mu b_x \);     \( y = y_0 + \lambda a_y + \mu b_y \);     \( z = z_0 + \lambda a_z + \mu b_z \)

We can emphasize that these parametric equations are dependent on the parameters \(\lambda\) and \(\mu\) by writing them as

\( x(\lambda, \mu) = x_0 + \lambda a_x + \mu b_x \);     \( y(\lambda, \mu) = y_0 + \lambda a_y + \mu b_y \);     \( z(\lambda, \mu) = z_0 + \lambda a_z + \mu b_z \)

Vector Function

We can use the above parametric equations to build a vector function to define a plane.

\( \vec{W}(\lambda, \mu) = (x_0 + \lambda a_x + \mu b_x)\hat{i} + (y_0 + \lambda a_y + \mu b_y)\hat{j} + (z_0 + \lambda a_z + \mu b_z)\hat{k} \)

More simply, we can write \( \vec{W}(\lambda, \mu) = x(\lambda, \mu)\hat{i} + y(\lambda, \mu)\hat{j} + z(\lambda, \mu)\hat{k} \)

Calculating The Angle Between Two Planes

Using the idea of the dot product, we can calculate the angle between two planes. All we need are two normal vectors, one for each plane. The angle between the normal vectors is the same as the angle between the planes. So we can use the equation

\( \displaystyle{ \cos(\theta) = \frac{| \vec{n}_1 \cdot \vec{n}_2 |}{\| \vec{n}_1 \| ~ \| \vec{n}_2 \|} }\)

Search 17Calculus

Practice 1

Find the equation of the plane containing the point \((2,3,-1)\) with normal vector \(\vec{n}=\langle 3,5,-2\rangle\).



Practice 2

Find the scalar equation of the plane with normal vector \( \vec{n} = \langle 2,-9,8 \rangle \) through the point \( (-1,4,-2) \).


Practice 3

Find the equation of the plane that goes through \( (3,-1,2) \), \( (8,2,4) \) and \( (-1,-2,-3) \).


Practice 4

Find the equation of the plane that passes through the points \( P(1,0,2) \), \( Q(-1,1,2) \) and \( R(5,0,3) \).


Practice 5

Find the scalar equation of the plane that goes through the points \(P(0,1,0)\), \(Q(2,-1,4) \), \( R(-3,5,-4) \).


Real Time Web Analytics
menu top search practice problems
menu top search practice problems 17