(lanedance said:the vector from the closest point of the line to the point will be perpindicular to the line (why?)
To find the closest point on a line to another point, you can use the formula for the shortest distance between a point and a line. This involves finding the perpendicular distance from the given point to the line, and then finding the point on the line that is closest to the given point.
The formula for finding the shortest distance between a point and a line is d = |ax0 + by0 + c| / √(a^2 + b^2), where (x0, y0) is the coordinates of the given point, and a, b, and c are the coefficients of the line's equation in standard form (ax + by + c = 0).
Yes, the formula for finding the shortest distance between a point and a line can be used to find the closest point on a line to a point that is not on the line. It will give you the coordinates of the point on the line that is closest to the given point.
The formula for finding the shortest distance between a point and a line applies to straight lines only. For curved lines, you can find the closest point by finding the tangent to the curve at the given point and then finding the intersection point of the tangent and the line.
Yes, there are other methods for finding the closest point on a line to another point, such as using vectors or linear algebra. These methods may be quicker and more efficient for certain types of problems, but the formula for finding the shortest distance between a point and a line is a commonly used approach that works for all types of lines.