First, we need to define the three points in our three-dimensional space: p(1,3,1), q(-4,7,-1), and r(6,-2,-1).
Next, we will calculate the two vectors in the plane of the points p, q, and r. These vectors are defined as follows:
- The vector pq, which we calculate as the difference between the coordinates of point q and point p. Therefore, pq = q - p.
- The vector pr, which we calculate as the difference between the coordinates of point r and point p. Therefore, pr = r - p.
After obtaining the two vectors, pq and pr, we can now find a vector that is orthogonal (or perpendicular) to them. This orthogonal vector will be perpendicular to the plane containing the points p, q, and r.
To find this orthogonal vector, it's important to know that the cross product of any two vectors in a plane gives a vector orthogonal to the plane. Therefore, we take the cross product of the vectors pq and pr.
The cross product of pq and pr gives us the vector that is orthogonal to the plane passing through the points p, q, and r.
After performing these calculations, we find that the orthogonal vector we are looking for is (-18, -20, 5). This is your final answer, a vector that is orthogonal to the plane through points p, q, and r.