## Calculus: Early Transcendentals 8th Edition

(a) $x+y+z=c$ This is the family of planes that have normal vectors parallel to $\lt 1,1,1\gt$. The distance from the plane to the origin varies by changing the value of $c$. For positive c values, the planes intersect to form an equilateral triangle in the first octant. For negative c values, the equilateral triangle is formed with the octant opposite the first. (b) $x+y+cz=1$ This is the family of planes that intercects the $xy$-plane at the 2D line $x+y=1$. With $c=0$, the plane is parallel to the z-axis. For larger $c$ values, the plane moves closer to the xy-plane. (c) $y cos \theta+zsin \theta=1$ Because there is no $x$, the planes will be parallel to the $x-$ axis. These lines are basically tangent lines to a circle of radius $1$. So, the planes are tangent lines to a circle of radius $1$ on the $yz$ plane extended in the direction parallel to the $x-axis$.
(a) $x+y+z=c$ This is the family of planes that have normal vectors parallel to $\lt 1,1,1\gt$. The distance from the plane to the origin varies by changing the value of $c$. For positive c values, the planes intersect to form an equilateral triangle in the first octant. For negative c values, the equilateral triangle is formed with the octant opposite the first. (b) $x+y+cz=1$ This is the family of planes that intercects the $xy$-plane at the 2D line $x+y=1$. With $c=0$, the plane is parallel to the z-axis. For larger $c$ values, the plane moves closer to the xy-plane. (c) $y cos \theta+zsin \theta=1$ Because there is no $x$, the planes will be parallel to the $x-$ axis. These lines are basically tangent lines to a circle of radius $1$. So, the planes are tangent lines to a circle of radius $1$ on the $yz$ plane extended in the direction parallel to the $x-axis$.