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: Using the distance/section approach instead:
Compute \(AB=\sqrt{(-1)^2+(-5)^2+7^2}=\sqrt{1+25+49}=\sqrt{75}=5\sqrt3\), and \(AC=\sqrt{1^2+5^2+(-7)^2}=\sqrt{75}=5\sqrt3\).

Step 2: Computing BC:
\(\vec{BC}=C-B=(2,10,-14)\), so \(BC=\sqrt{4+100+196}=\sqrt{300}=10\sqrt3\).

Step 3: Verifying the sum condition for collinearity:
\(AB+AC=5\sqrt3+5\sqrt3=10\sqrt3=BC\), so \(A\) lies exactly between \(B\) and \(C\) on the same line.

Final Answer:
\[ \boxed{A,B,C \text{ are collinear (}AB+AC=BC\text{)}} \]