Final answer:
To find the two points on the ellipsoid where the tangent plane is normal to the vector v = <1, 1, -2>, we need to solve the equations of the ellipsoid and the tangent plane simultaneously.
Step-by-step explanation:
To find the two points on the ellipsoid where the tangent plane is normal to the vector v = <1, 1, -2>, we need to solve two equations simultaneously. The first equation is the equation of the ellipsoid: x²/4 + y²/9 + z² = 1. The second equation is the equation of the tangent plane, which has the form: a(x - x₁) + b(y - y₁) + c(z - z₁) = 0. We know that the normal vector of the tangent plane is n = <a, b, c>. Since we want the tangent plane to be normal to v = <1, 1, -2>, we can set n = v. By solving the two equations, we can find the two points.
Let's solve the equations:
Equation of the ellipsoid: x²/4 + y²/9 + z² = 1
Equation of the tangent plane: a(x - x₁) + b(y - y₁) + c(z - z₁) = 0
By substituting the values of a = 1, b = 1, c = -2 into the equation of the tangent plane and solving simultaneously with the equation of the ellipsoid, we can find the two points where the tangent plane is normal to v = <1, 1, -2>.