Question

In: Math

a. consider the plane with equation -x+y-z=2, and let p be the point (3,2,1)in R^3. find...

a. consider the plane with equation -x+y-z=2, and let p be the point (3,2,1)in R^3. find the distance from P to the plane.

b. let P be the plane with normal vector n (1,-3,2) which passes through the point(1,1,1). find the point in the plane which is closest to (2,2,3)

Solutions

Expert Solution

a. The distnace of the point from the plane is

here we have plane ,

point ,

we get distance as

b. we have normal vector on plane as

and it passes through

we get equation of the plane as

we find a line containing the point and direction ratio same as normal vector ,

plugging this into plane equation we get

plugging this value of t in line equation we get

so the closest point on plane is


Related Solutions

Find the equation of the tangent plane to the surface x + y^2 + z^3 +...
Find the equation of the tangent plane to the surface x + y^2 + z^3 + sin(x − yz) = 7 at the point (2, 2, 1).
Find the coordinates of the point (x, y, z) on the plane z = 4 x...
Find the coordinates of the point (x, y, z) on the plane z = 4 x + 1 y + 4 which is closest to the origin.
*(1)(a) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z):...
*(1)(a) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=c}. (b) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): x=a}. (c) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): y=b}. *(2) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=kx+b} assuming both b and k are positive. (a) For what value of...
find the point lying on the intersection of the plane, x + (1/4)y + (1/3)z =...
find the point lying on the intersection of the plane, x + (1/4)y + (1/3)z = 0 and the sphere x 2 + y 2 + z 2 = 25 with the largest z-coordinate. (x,y,z)=(_)
6) a). Find the equation of the plane through the origin and perpendicular to x+y+z =...
6) a). Find the equation of the plane through the origin and perpendicular to x+y+z = 5 and 2x+y−2z = 7 b). Let A = (−1,3,0), B = (3,2,4) and C = (1,−1,5). ( I ) Find an equation for the plane that passes through these three points. ( II ) Find the area of the triangle determined by these three points.
Find the point of intersection of the lines (x-2)/- 3 = (y-2)/6 = z-3 & (x-3)/2...
Find the point of intersection of the lines (x-2)/- 3 = (y-2)/6 = z-3 & (x-3)/2 = y+5 = (z+2)/4. Write the answer as (a, b, c). If they are not cut, write: NO
Find the equation of the plane through the point (1,1,1) which is perpendicular to the line of intersection of the two planes x−y−3z=−1 and x−3y+z= 2.
Find the equation of the plane through the point (1,1,1) which is perpendicular to the line of intersection of the two planes x−y−3z=−1  and x−3y+z= 2.
Let F(x, y, z) = z tan^−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find the...
Let F(x, y, z) = z tan^−1(y^2)i + z^3 ln(x^2 + 7)j + zk. Find the flux of F across S, the part of the paraboloid x2 + y2 + z = 29 that lies above the plane z = 4 and is oriented upward.
Find an equation of the tangent plane to the surface x5+5z2ey−x=848 at point P=(3, 4, 11e√)....
Find an equation of the tangent plane to the surface x5+5z2ey−x=848 at point P=(3, 4, 11e√). (Use symbolic notation and fractions where needed. Your answer should be in the form ax+by+cz=1.)
Given the function u(p,q,r)=((p-q)/(q-r)), with p=x+y+z,q=x-y+z, and r=x+y-z, find the partial derivatives au/ax=, au/ay=, au/az=
Given the function u(p,q,r)=((p-q)/(q-r)), with p=x+y+z,q=x-y+z, and r=x+y-z, find the partial derivatives au/ax=, au/ay=, au/az=
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT