Using the Cross Product to Find the Equation of a Plane We know from Property (f) that the cross product u X v is orthogonal to both of the vectors u and v. Therefore, if u and v are a pair of direction vectors for a plane, then u X v is a normal vector for the plane. An example similar to, yet different from, the one given below can be found in the video Determinants: Properties and Applications 2 starting at 7:40. = Example: Consider the plane P in R³ that contains the three points p₁ = [1,0, 1], p2 = [1, 0, 0] and p3 Let u = P2 P₁ = and let v = P3 P1 = Since u and v (No answer given) scalar multiples of each other, they (No answer given) parallel and hence u and v serve as direction vectors for P. Using the cross product of the vectors u and v, we obtain that a normal vector for this plane is n=uxv= [0, 1, 0].
Using the Cross Product to Find the Equation of a Plane We know from Property (f) that the cross product u X v is orthogonal to both of the vectors u and v. Therefore, if u and v are a pair of direction vectors for a plane, then u X v is a normal vector for the plane. An example similar to, yet different from, the one given below can be found in the video Determinants: Properties and Applications 2 starting at 7:40. = Example: Consider the plane P in R³ that contains the three points p₁ = [1,0, 1], p2 = [1, 0, 0] and p3 Let u = P2 P₁ = and let v = P3 P1 = Since u and v (No answer given) scalar multiples of each other, they (No answer given) parallel and hence u and v serve as direction vectors for P. Using the cross product of the vectors u and v, we obtain that a normal vector for this plane is n=uxv= [0, 1, 0].
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.2: Determininants And Elementary Operations
Problem 48E
Related questions
Question
![Using the Cross Product to Find the Equation of a Plane
We know from Property (f) that the cross product u X v is orthogonal to both of the vectors u and v. Therefore, if u and
v are a pair of direction vectors for a plane, then u X v is a normal vector for the plane.
An example similar to, yet different from, the one given below can be found in the video Determinants: Properties and
Applications 2 starting at 7:40.
=
Example: Consider the plane P in R³ that contains the three points p₁ = [1,0, 1], P₂ = [1, 0, 0] and p3
Let u = P2 P₁ =
-[||▬▬▬] and let v P3-P₁ -[|||
= P1
Since u and v (No answer given) scalar multiples of each other, they (No answer given) parallel
and hence u and v serve as direction vectors for P.
Using the cross product of the vectors u and v, we obtain that a normal vector for this plane is
n=uxv=
[0, 1, 0].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98fdc1e3-416c-49e2-9808-872164c20a53%2F325ca857-ccc3-4b10-a5ff-36f656483048%2Fy382bw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Using the Cross Product to Find the Equation of a Plane
We know from Property (f) that the cross product u X v is orthogonal to both of the vectors u and v. Therefore, if u and
v are a pair of direction vectors for a plane, then u X v is a normal vector for the plane.
An example similar to, yet different from, the one given below can be found in the video Determinants: Properties and
Applications 2 starting at 7:40.
=
Example: Consider the plane P in R³ that contains the three points p₁ = [1,0, 1], P₂ = [1, 0, 0] and p3
Let u = P2 P₁ =
-[||▬▬▬] and let v P3-P₁ -[|||
= P1
Since u and v (No answer given) scalar multiples of each other, they (No answer given) parallel
and hence u and v serve as direction vectors for P.
Using the cross product of the vectors u and v, we obtain that a normal vector for this plane is
n=uxv=
[0, 1, 0].
Transcribed Image Text:Now, using this and the fact that the point p₁ is in the plane we obtain that the general equation for this plane is
(Use x, y, z as the variables in your computation for this equation.)
There is yet another way to use the determinant to obtain the general equation of a plane in R³. It essentially amounts to
the same calculation as with the cross product above, but the argument is different: see the first half of the video
Determinants: Properties and Applications 2.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning