Define F(x, y, z) = x+4y+z, then 2x+8y+2z the gradient of F(x, y, z) is VF(x,...

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Define F(x, y, z) = x+4y+z, then 2x+8y+2z the gradient of F(x, y, z) is VF(x, y, z) = ( > (input functions, separate by a comma) Define an ellipsoid via F(x, y, z), which is x +4y + z = 18. (a) Find an equation of the tangent plane to the ellipsoid at the point (1, 2, -1). First, you need to know that the normal direction of this tangent plane must be parallel to the vector (2, b, c), where b = O 0 If the tangent plane is 2x+by+cz=d, then d can only be (b) Find parametric equations of the line that is normal to the ellipsoid at the point (1,2, -1). 16 -2 c= 5 40 First, you need to know that the direction of this normal line must be parallel to the vector (p,q,-5), where p: If z=-1-5t is one of the parametric equations of this normal line, then the other two equations are 8 0 X (no answer) y (input the constant term before the t term, e.g., 8t + 5 should be input as 5+8t. This agrees with the "increasing power order" we have been using, since 8t + 5 is 5t + 8t) Define F(x, y, z) = x+4y+z, then 2x+8y+2z the gradient of F(x, y, z) is VF(x, y, z) = ( > (input functions, separate by a comma) Define an ellipsoid via F(x, y, z), which is x +4y + z = 18. (a) Find an equation of the tangent plane to the ellipsoid at the point (1, 2, -1). First, you need to know that the normal direction of this tangent plane must be parallel to the vector (2, b, c), where b = O 0 If the tangent plane is 2x+by+cz=d, then d can only be (b) Find parametric equations of the line that is normal to the ellipsoid at the point (1,2, -1). 16 -2 c= 5 40 First, you need to know that the direction of this normal line must be parallel to the vector (p,q,-5), where p: If z=-1-5t is one of the parametric equations of this normal line, then the other two equations are 8 0 X (no answer) y (input the constant term before the t term, e.g., 8t + 5 should be input as 5+8t. This agrees with the "increasing power order" we have been using, since 8t + 5 is 5t + 8t)