Finding the foot of the perpendicular from a point to a line in 3D

Hi Tanishka, let's solve this three-dimensional geometry problem together. We are given the origin point O, point A, and point B. We need to find the foot of the perpendicular from B to the line OA.

Let's visualize this scenario. Here is the line passing through the origin O and the point A. We let M with coordinates x, y, z be the foot of the perpendicular drawn from point B to the line OA.

Let's start by finding the equation of the line OA. First, the direction ratios of the line OA are simply the differences of the coordinates of A and the origin O.

Since the line passes through the origin and has these direction ratios, we can write its symmetric equation as x over one equals y over two equals z over three.

Let's set this equal to a scalar constant, k. This allows us to write any general point M on the line in terms of k.

By equating each term to k, we find that the coordinates of M are k, two k, and three k.

Now, let's find the direction ratios of the line segment BM. We subtract the coordinates of B from the coordinates of M.

Since BM is perpendicular to OA, the dot product of their direction ratios must equal zero.