The formula for calculating the distance between a point and a plane in vector calculus is given by:
d = |ax0 + by0 + cz0 + d| / √(a² + b² + c²), where (x0, y0, z0) is the coordinates of the point and ax + by + cz + d = 0 is the equation of the plane.
To find the coordinates of the point on the plane closest to a given point, first calculate the distance between the given point and the plane using the formula d = |ax0 + by0 + cz0 + d| / √(a² + b² + c²). Then, substitute this distance into the equation of the plane to solve for the coordinates of the closest point.
No, the distance between a point and a plane cannot be negative. Distance is a measure of length and is always positive.
The absolute value in the distance formula ensures that the result is always positive, as distance is a measure of length and cannot be negative. It also accounts for the possibility of the point being on the opposite side of the plane from its normal vector.
Yes, the distance between a point and a plane can be zero if the given point lies on the plane. In this case, the point is said to be coplanar with the plane.