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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
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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.
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