Find the point on the plane x y z 2 that is closest to the point 2 7 8


Introduction

In this article, we will be focusing on how to find the point on the plane x y z 2 that is closest to the point 2 7 8. We will be using the definition of a vector to help us in our calculations. A vector is a mathematical object that has both a magnitude and a direction. It is often represented by an arrow.

The point on the plane that is closest to the point

The point on the plane that is closest to the point (2,7,8) is the point (2,7,2).

How to find the point on the plane that is closest to the point

To find the point on the plane that is closest to the point, we need to find the normal vector of the plane. The normal vector is perpendicular to the plane. To find the normal vector, we can use the cross product:

Next, we need to find a point on the plane. Any point on the plane will do. We’ll call this point P.

Now that we have a point and a normal vector, we can use these to find the equation of the plane. The equation of a plane is:

Now that we have the equation of the plane, we can plug in our values for P and n and solve for t. This will give us the distance from our point to the closest point on the plane.

t = (2 – 2)/2 + (7 – 8)/(-1) + (8 – 2)/3 = 1/5

The closest point on the plane to our original point is:

P’ = P + tn = (2, 7, 8) + (1/5)(2, -1, 3) = (2 + 2/5, 7 – 1/5, 8 + 3/5) = (4/5, 6 4/5, 11/5)

Conclusion

The point on the plane that is closest to the point (2,7,8) is the point (2,7,2).


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