Question

Show that the points A(1,2,7),B(2,6,3) and C(3,10,-1) are collinear.

Answer 1

To show that three points are collinear, we can check if the vectors formed by connecting them are parallel. In this case, by calculating the cross product of the vectors AB and BC, we find that it is the zero vector, indicating that the points are collinear.

Step-by-step explanation:

To show that the points A(1,2,7), B(2,6,3), and C(3,10,-1) are collinear, we need to check if the vectors AB and BC are parallel.

1. Calculate the vector AB by subtracting the coordinates of point A from the coordinates of point B: AB = (2-1, 6-2, 3-7) = (1, 4, -4).
2. Calculate the vector BC by subtracting the coordinates of point B from the coordinates of point C: BC = (3-2, 10-6, -1-3) = (1, 4, -4).
3. Check if the vectors AB and BC are parallel. If the cross product of AB and BC is the zero vector, then they are parallel. Calculate the cross product: AB x BC = (4*(-4) - (-4)*4, -4*1 - 1*4, 1*4 - 1*4) = (0, 0, 0).
4. Since the cross product is the zero vector, it means that AB and BC are parallel, and therefore, the points A, B, and C are collinear.