# Recitation 3

Section 10.5 Problem 10: Find the parametric equation and symmetric equation for the line of intersection of the planes $x + 2y + 3z = 1$ and $x + y + z = 1$.

Comment: To find the parametric equation and symmetric equation of the line, it is enough to find a point $(x_0,y_0,z_0)$ on the line and the direction $\langle A,B,C\rangle$ of the line. Then the parametric equation is $x(t)=x_0+At, y(t)=y_0+Bt, z(t)=z_0+Ct$ and the symmetric equation is $\frac{x-x_0}{A}=\frac{y-y_0}{B}=\frac{z-z_0}{C}$ unless $ABC=0$. For this problem, it is easy to observe that $(1,0,0)$ is on the line and the cross product of the normal vectors of the planes is a direction of the line.

Section 10.5 Problem 27: Find the equation of the plane that passes through the point $(6, 0, 2)$ and contains the line $x = 4 + 2t, y = 3 + 5t, z = 7 + 4t$.

Comment: The point $(6,0,2)$ is on the plane for free. It is easy to see the point $(4,3,7)$ is also on the plane since it is on the line. So the vector between those two points $\langle -2,3,5 \rangle$ is parallel to the plane. Also the direction of the line $\langle 2, 5, 4 \rangle$ is also parallel to the plane. Thus their cross product is a normal vector of the plane.

Section 10.5 Problem 45: Which of the following four planes are parallel? Are any of them identical? $P1: 3x+6y-3z=6, P2: 4x-12y+8z=5, P3: 9y=1+3x+6z, P4: z= x+2y-2$.

Comment: Compare the normal vectors of the planes to see whether they are parallel.

Section 10.5 Problem 49: Find the distance from the point to the given plane. $(1,-2,4), 3x+2y+6z=5$.

Comment: Use the formula for the distance from a point to a plane.

Section 10.5 Problem 51: Find the distance between the given parallel planes. $2x-3y+z=4, 4x-6y+2z=3$.

Comment: It is enough to find the distance from a point on the first plane to the second plane.

Section 10.6 Problem 7: Describe and sketch the surface. $xy=1$

Comment: Draw the curve in the x-y plane and use it to form a cylinder.

Section 10.6 Problem 18: Use traces to sketch and identify the surface. $4x^2 -16y^2 +z^2 =16$.

Comment: The x-sections and the z-sections are hyperbolas and the y-sections are ellipses. Thus the surface is hyperboloid with one sheet.

Section 10.6 Problem 25: Reduce the equation to one of the standard forms, classify the surface, and sketch it. $4x^2 +y^2 +4z^2 -4y-24z+36=0$.

Comment: Complete the squares and see it is an ellipsoid.

Section 10.6 Problem 31: Find an equation for the surface consisting of all points that are equidistant from the point $(-1, 0, 0)$ and the plane $x = 1$. Identify the surface.

Solution: Let $P=(x,y,z)$ be a point on the surface. Then $(x+1)^2+y^2+z^2=(x-1)^2$. Notice that the left hand side is the square of the distance from $P$ to the point $(-1, 0, 0)$ and the right hand side is the square of the distance form $P$ to the plane $x = 1$. This equation gives us $2x+y^2+z^2=0$ which is an elliptic paraboloid.