Question

Show that the points \( A(2, 3, -4), B(1, -2, 3) \) and \( C(3, 8, -11) \) are collinear.

Hint:

To check if three points are collinear, find the vectors between two pairs of points and verify if one vector is a scalar multiple of the other.

Answer 1

Step 1: To show that the points are collinear, find the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \): \[ \overrightarrow{AB} = B - A = (1 - 2, -2 - 3, 3 + 4) = (-1, -5, 7) \] \[ \overrightarrow{AC} = C - A = (3 - 2, 8 - 3, -11 + 4) = (1, 5, -7) \] Step 2: Check if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are scalar multiples of each other. We observe that: \[ \overrightarrow{AC} = -\overrightarrow{AB} \] Step 3: Since \( \overrightarrow{AC} \) is a scalar multiple of \( \overrightarrow{AB} \), the points \( A, B, C \) are collinear. Thus, the points are collinear.