Question

The unit normal vector to the level surface $$ x^2 y + 2 x z = 4 $$ at the point $ (2, -2, 3) $ is:

• A. \( \frac{i - 2j + 2k}{3} \)
• B. \( \frac{i - 2j - 2k}{3} \)
• C. \( \frac{-i + 2j + 2k}{3} \)
• D. \( \frac{i + 2j - 2k}{3} \)

Hint:

Normal to surface given by gradient vector \( \nabla F \); normalize to get unit normal.

Answer 1

The Correct Option is C

The gradient vector \( \nabla F \) at point gives the normal to the surface: \[ F(x,y,z) = x^2 y + 2 x z - 4 = 0 \] Calculate partial derivatives: \[ \frac{\partial F}{\partial x} = 2 x y + 2 z, \quad \frac{\partial F}{\partial y} = x^2, \quad \frac{\partial F}{\partial z} = 2 x \] At point \( (2, -2, 3) \): \[ \nabla F = \left( 2 \cdot 2 \cdot (-2) + 2 \cdot 3 \right) i + (2^2) j + (2 \cdot 2) k = (-8 + 6) i + 4 j + 4 k = -2 i + 4 j + 4 k \] Unit vector: \[ \frac{-2 i + 4 j + 4 k}{\sqrt{(-2)^2 + 4^2 + 4^2}} = \frac{-2 i + 4 j + 4 k}{\sqrt{4 + 16 + 16}} = \frac{-2 i + 4 j + 4 k}{\sqrt{36}} = \frac{-2 i + 4 j + 4 k}{6} = \frac{-i + 2 j + 2 k}{3} \]