Q.Show that the points $(2, 3, 4)$, $(-1, -2, 1)$, $(5, 8, 7)$ are collinear.

Concept: Collinearity Condition — three points are collinear if the vectors formed by them are linearly dependent, i.e., one is a scalar multiple of another.

Step 1: Let $A(2,3,4)$, $B(-1,-2,1)$, $C(5,8,7)$. Find vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$.

Step 2: Check if $\overrightarrow{BC}$ is a scalar multiple of $\overrightarrow{AB}$.

$\frac{6}{-3} = -2,\quad \frac{10}{-5} = -2,\quad \frac{6}{-3} = -2$ …

Why the Collinearity Condition Works

Three points are collinear if they lie on the same straight line. The cleanest way to check this in 3D geometry is to use vectors: pick one point as a reference, then form vectors to the other two points. If those two vectors are parallel (one is a scalar multiple of the other), the three points must be collinear.

Why? Because if you stand at point A and look toward B, and also toward C, and both directions are exactly the same (or exactly opposite), then B and C lie on the same line through A. The scalar multiple tells you how far apart they are along that line.

Let's apply this to the given points.

1. Label the points and pick a reference

Let $A = (2, 3, 4)$, $B = (-1, -2, 1)$, $C = (5, 8, 7)$. We'll use $A$ as our reference point.

2. Form the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$

$\overrightarrow{AB} = B - A = (-1 - 2,\; -2 - 3,\; 1 - 4) = (-3, -5, -3)$

$\overrightarrow{AC} = C - A = (5 - 2,\; 8 - 3,\; 7 - 4) = (3, 5, 3)$

3. Check if the vectors are parallel

Two vectors $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are parallel if their corresponding components are proportional:

provided no denominator is zero. Here:

All three ratios equal $-1$. So $\overrightarrow{AB} = -1 \cdot \overrightarrow{AC}$. …

Method: Test collinearity with a shared-point direction check

To prove three points $A,B,C$ lie on one straight line, show that two vectors built from them — sharing a common point — are parallel. Parallel direction plus a shared point forces a single line.

Steps

Step 1: Pick one point as the anchor (say $A$) and form two vectors from it:

Using the same anchor for both is essential — that shared point is what upgrades "parallel" to "collinear."

Step 2: Test for parallelism. Check whether one vector is a scalar multiple of the other, i.e. the components are proportional:

Equivalently, $\vec{AB}\times\vec{AC}=\vec0$. …

Common Mistakes

Mistake 1: Checking that $\vec{AB}$ and $\vec{BC}$ have equal length instead of equal direction.

Why it's wrong: collinearity is about direction, not magnitude — equal lengths in different directions are not collinear. Correct approach: test proportionality of components (one vector a scalar multiple of the other).

Mistake 2: Concluding "not collinear" because the scalar is negative. …