Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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].

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
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].
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].
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.
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
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 5 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS