Find the equation for the plane through the points Po(-1,-5,-2), Qo(3,-4,2), and Ro(1,1,5). ..... The equation of the plane is

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the equation for the plane through the points Po(1,1.1), Qo(2,5.2), and Ro(3,- 2,-6).
Find the vectors PoQ, and P,Ro- Subtract the coordinates of Po(1.1,1) from Qo(2,5,2).
(1, 4. 1)
Repeat the previous step for Ro(3. -2, - 6) and Po(1,1,1).
PORO =
(2. - 3. - 7)
Take the cross product of the two vectors found above. Start by filling out the determinant below.
k
PoQ, x P,R, =
2 - 3 - 7
Take the determinant and find the equation for the normal vector, n.
n = -25i+ 9j - 11k
The standard equation for a plane through the point P,(Xo Yo Zo) and perpendicular to a vector n= (A, B. C) is
Ax + By + Cz = Ax, + By, + Cz,. Substitute the appropriate values into this equation using n= (-25, 9, - 11)
and Po(1,1,1).
Ax + By + Cz = Axg +By, + Cz,
(- 25)x + (9)y + (-11)z = (-25)(1) + (9)(1) + (- 11)(1)
Finally, simplify and write the equation of the plane in the form Ax + By + Cz = D.
- 25x + 9y - 11z = -27
Transcribed Image Text:Find the equation for the plane through the points Po(1,1.1), Qo(2,5.2), and Ro(3,- 2,-6). Find the vectors PoQ, and P,Ro- Subtract the coordinates of Po(1.1,1) from Qo(2,5,2). (1, 4. 1) Repeat the previous step for Ro(3. -2, - 6) and Po(1,1,1). PORO = (2. - 3. - 7) Take the cross product of the two vectors found above. Start by filling out the determinant below. k PoQ, x P,R, = 2 - 3 - 7 Take the determinant and find the equation for the normal vector, n. n = -25i+ 9j - 11k The standard equation for a plane through the point P,(Xo Yo Zo) and perpendicular to a vector n= (A, B. C) is Ax + By + Cz = Ax, + By, + Cz,. Substitute the appropriate values into this equation using n= (-25, 9, - 11) and Po(1,1,1). Ax + By + Cz = Axg +By, + Cz, (- 25)x + (9)y + (-11)z = (-25)(1) + (9)(1) + (- 11)(1) Finally, simplify and write the equation of the plane in the form Ax + By + Cz = D. - 25x + 9y - 11z = -27
Find the equation for the plane through the points Po(-1,-5,-2), Qo(3,-4,2), and Ro(1,1,5).
The equation of the plane is
Transcribed Image Text:Find the equation for the plane through the points Po(-1,-5,-2), Qo(3,-4,2), and Ro(1,1,5). The equation of the plane is
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