What point on the plane x - y + z = 3 is closest to the point (2,3,1)? The close

Question

What point on the plane x - y + z = 3 is closest to the point (2,3,1)? The closest point on the plane is (Simplify your answers.) What point on the plane x - y + z = 1 is closest to the point (3,1,3)? The closest point on the plane is (Simplify your answers.) Compute the directional derivative of the following function at the given point P in the direction of the given vector. Be sure to use a unit vector for the direction vector. The directional derivative is (Type an exact answer, using radicals as needed.)

Explanation / Answer

n = <1, -1, 1>

The closest point in the plane to the given point P(2,3,1) is on the line thru P and perpendicular to the plane. Since the line is perpendicular to the plane, the normal vector of the plane is also the directional vector of the line. The equation of the line is:

L(t) = P + tn
L(t) = <2,3,1> + t<1, -1, 1>
where parameter t ranges over the real numbers

The desired point Q is the point of intersection between the given plane and the line. Substitute the parametric values for x, y, and z into the equation of the line and solve for t.

x - y + z = 3
(2 + t) - (3 - t) + (1 + t) = 3
3t = 3
t=1

L(t):
x = 2 + t = 2+ 1 = 3
y = 3 - t = 3-1 = 2
z = 1 + t = 1+1 = 2

The point of intersection is Q(3,2,2). This point is the closest point in the plane to P.

the same goes for number 2 same steps just plug in the different numbers

and 3, the directional derivative is The gradient defined at the point given dotted by the direction

so the gradient is the partial derivatives of root 4-x^(2)-2y is

((-x/root(4-x^2-2y),-1/root(4-x^(2)-2y)) evaluate this at 2,-2 yields ( -2/root8, -1/root8)

now dot it with the direction vector: yields -2/root8 x 1/root10 + -1/root8 x 3/root10

= -2/root80 +-3/root80 = -5/root80

finite

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