In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. a.2x+3y+4z-12=0 b.3y+4z-6=0 c. x+y+z=1 d. 5y+8=0

Find the distance of the plane 2x-y-2z-9=0 from the origin.

In the follwing cases , determine whether the given planes are parallel or perpendicular and in case they are neither , find the angles between them. a. 7x+5y+6z+30=0 and 3x-y-10z+4=0 b. 2x+y+3z-2=0 and x-2y+5=0 c. 2x-2y +4z +5=0 and 3x-3y+6z-1=0 d. 2x-y+3z-1=0 and 2x-y+3z+3=0 e. 4x+8y+z-8=0 and y+z-4=0

In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree. g(x, y, z) = (sqrt(3x^(2) + 5y^(2) + z^(2)))/(4x + 7y)

In each of the following cases , determine whether the following function is homogeneous or not. If it is so , find the degree. (i) f(x,y)=x^2y+6x^3+7 (ii) h(x,y)=(6x^2y^3-piy^5+9x^4y)/(2020x^2+2019y^2) (iii) g(x,y,z)=(sqrt(3x^2+5y^2+z^2))/(4x+7y) (iv) U(x,y,z) =xy+sin((y^2-2z^2)/(xy))

In the following cases, find the distance of each of the given points from the corresponding given plane. {:("Point","Plane"),("a. (0,0,0),3x-4y+12z=3),(b. (3,-2,1),2x-y+2z+3=0),(c.(2,3,-5),x+2y-2z=9),(d.(-6,0,0),2x-3y+6z-2=0):}

Find the distance between the parallel planes x + 2y - 2z + 1 = 0 and 2x + 4y - 4z + 5 = 0.