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

Three points are collinear when one displacement vector is a scalar multiple of another (same direction, shared point).

Displacements from $A$.

$\vec{AC} = C-A = (3-2,\,8-3,\,-11-(-4)) = (1,5,-7).$ …

The idea

Three points are collinear if they lie on one straight line. In vector terms, the displacements from a common point must be parallel — one a scalar multiple of the other. If $\vec{AB}$ and $\vec{AC}$ are parallel (and both start at $A$), then $B$ and $C$ lie on the same line through $A$.

Find the displacement vectors

Use $A$ as the reference point.

Check for a scalar multiple

Compare component by component:

All three ratios are equal to $-1$, so

Conclude …

Method: Testing Whether Three Points Are Collinear

Use this when you must decide whether three points $A,B,C$ lie on one straight line.

Steps

Step 1: Form two displacement vectors from a shared point.

Compute, say, $\vec{AB}=B-A$ and $\vec{AC}=C-A$ (both starting at $A$). Using a common starting point is what guarantees that parallel directions force all three points onto one line.

Step 2: Test whether one is a scalar multiple of the other. …

Common Mistakes

Mistake 1: Concluding non-collinearity because the two vectors point in opposite senses.

Why it's wrong: here $\vec{AC}=-\vec{AB}$; a negative scalar multiple is still parallel, so the points are collinear. Correct approach: collinearity only needs $\vec{AC}=\lambda\vec{AB}$ for some scalar, positive or negative.

Mistake 2: Checking only that the segments have equal length. …