Question

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

Hint:

Compute AB and AC and check if one is a scalar multiple of the other.

Answer 1

Step 1: Key Approach:
Three points are collinear if the vectors joining them are parallel (scalar multiples of each other).

Step 2: Computing the vectors:
\(\vec{AB}=B-A=(1-2,-2-3,3-(-4))=(-1,-5,7)\). \(\vec{AC}=C-A=(3-2,8-3,-11-(-4))=(1,5,-7)\).

Step 3: Comparing the vectors:
\(\vec{AC}=(1,5,-7)=-1\times(-1,-5,7)=-\vec{AB}\).

Final Answer:
Since \(\vec{AC}\) is a scalar multiple of \(\vec{AB}\), the points \(A\), \(B\), \(C\) are collinear.\[ \boxed{A,\,B,\,C\text{ are collinear}} \]