Find a point on the surface $x y z = 8$ such that the normal vector at the point is parallel to the vector $\{ 4,2,1 \}$ .
A) $( 1,2 , - 4 )$
B) $( - 1 , - 2,4 )$
C) $( 1 , - 2 , - 4 )$
D) $( - 1 , - 2 , - 4 )$
E) $( - 1,2,4 )$
F) $( - 1,2 , - 4 )$
G) $( 1 , - 2 , - 4 )$.
H) $( 1,2,4 )$

Question

Find a point on the surface $x y z = 8$ such that the normal vector at the point is parallel to the vector $\{ 4,2,1 \}$ .
A) $( 1,2 , - 4 )$ 
B) $( - 1 , - 2,4 )$ 
C) $( 1 , - 2 , - 4 )$ 
D) $( - 1 , - 2 , - 4 )$
E) $( - 1,2,4 )$  
F) $( - 1,2 , - 4 )$ 
G) $( 1 , - 2 , - 4 )$.
H) $( 1,2,4 )$

Quizplus · Accepted Answer

The gradient of the surface $x y z = 8$ is $\nabla(xyz) = \{yz, xz, xy\}$. For the normal vector to be parallel to $\{4,2,1\}$, $\nabla(xyz)$ must be a scalar multiple of $\{4,2,1\}$. At point $(1,2,4)$, $\nabla(xyz) = \{8, 4, 2\}$, which is $2$ times $\{4,2,1\}$.

Find a Point on the Surface xyz=8x y z = 8xyz=8

Find a Point on the Surface $x y z = 8$