Q.Show that the points A$(1, 2, 7),$ B$(2, 6, 3)$ and C$(3, 10, -1)$ are collinear.

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

Step 1: Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$.

Step 2: Compare the two vectors.

We see $\overrightarrow{BC} = 1 \cdot \overrightarrow{AB}$, i.e., they are scalar multiples (here the scalar is $1$). …

Why This Works: The Collinearity Condition

Three points are collinear if they lie on a single straight line. The cleanest way to check this in 3D geometry is to use vectors. If points A, B, and C are collinear, then the vector from A to B and the vector from B to C must point in the same (or exactly opposite) direction. In mathematical terms, one vector must be a scalar multiple of the other.

Think of it like walking: if you go from A to B, and then from B to C, and you never change direction (you just keep walking straight), then all three points are on the same line. The vector approach captures this perfectly.

Let's apply this step by step.

1. Find the vector $\overrightarrow{AB}$.

The vector from A to B is found by subtracting the coordinates of A from B:

2. Find the vector $\overrightarrow{BC}$.

Similarly, the vector from B to C is:

3. Compare the two vectors.

Look at what we have:

They are identical! That means $\overrightarrow{AB} = 1 \cdot \overrightarrow{BC}$. The scalar $\lambda$ here is exactly $1$. …

Method: Testing Collinearity of Three Points

Use this to decide whether three given points lie on one straight line.

Steps

Step 1: Form two direction vectors that share a common point.

Compute $\vec{AB}=B-A$ and $\vec{BC}=C-B$. Sharing the point $B$ is what forces all three points onto one line if the directions match.

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

$A,B,C\text{ collinear} \iff \vec{AB}=\lambda\,\vec{BC} \iff \vec{AB}\times\vec{BC}=\vec{0}$ …

Common Mistakes

Mistake 1: Checking equal lengths instead of equal direction.

Why it's wrong: collinearity is about direction — two segments can be equal in length yet not aligned. Correct approach: show $\vec{AB}$ is a scalar multiple of $\vec{BC}$ (equivalently $\vec{AB}\times\vec{BC}=\vec{0}$). …